Introduction to Trigonometric Identities: Videos & Practice Problems

Understanding the even and odd identities of trigonometric functions is crucial for simplifying expressions. The cosine function is even, meaning cos(-θ)=cos(θ), while the sine function is odd, so sin(-θ) = -sin(θ). Pythagorean identities, such as sin²(θ) + cos²(θ) = 1, help in simplifying complex trigonometric expressions and verifying identities effectively.

Even and Odd Identities

Understanding the classification of trigonometric functions as even or odd is essential for simplifying expressions and solving problems effectively. An even function, such as cosine, satisfies the property that f(-x) = f(x). This means that the graph of an even function is symmetric about the y-axis. For example, the cosine function evaluated at \(\frac{\pi}{2}\) yields f\left(\frac{\pi}{2}\right) = 0 and f\left(-\frac{\pi}{2}\right) = 0, confirming its even nature.

In contrast, odd functions like sine exhibit the property f(-x) = -f(x), indicating symmetry about the origin. For instance, evaluating the sine function at \(\frac{\pi}{2}\) gives f\left(\frac{\pi}{2}\right) = 1 and f\left(-\frac{\pi}{2}\right) = -1, demonstrating that sine is odd.

Other trigonometric functions can also be classified based on their relationship to sine and cosine. The secant function, being the reciprocal of cosine, is even, while cosecant, as the reciprocal of sine, is odd. Tangent, defined as the ratio of sine to cosine, inherits the odd property, as does cotangent.

This classification leads to the formulation of even-odd identities, which are equations valid for all values of the variable involved. For example, the cosine of a negative angle can be expressed as \(\cos(-\theta) = \cos(\theta)\), while the sine and tangent of a negative angle are given by \(\sin(-\theta) = -\sin(\theta)\) and \(\tan(-\theta) = -\tan(\theta)\), respectively.

When applying these identities, it is crucial to recognize when the argument of the trigonometric function is negative. This awareness allows for the appropriate use of even-odd identities to simplify calculations. For instance, to evaluate \(\csc(-\frac{\pi}{6})\), one can rewrite it as \(\frac{1}{\sin(-\frac{\pi}{6})}\), and using the identity for sine, this simplifies to \(-\frac{1}{\frac{1}{2}} = -2\).

In summary, mastering the even-odd identities and their applications is vital for efficiently solving trigonometric problems, particularly when dealing with negative angles.

Example 1

Understanding even and odd identities is crucial for simplifying trigonometric expressions, especially when dealing with negative arguments. These identities allow us to rewrite expressions in a more manageable form, often eliminating negative angles entirely.

For instance, consider the expression involving the negative tangent of negative theta, expressed as -tan(-\theta). According to the even-odd identity for tangent, we know that tan(-\theta) = -tan(\theta). By substituting this identity into our expression, we have:

-tan(-\theta) = -(-tan(\theta)) = tan(\theta).

This simplification shows that two negatives result in a positive, allowing us to express the original function without any negative arguments.

Next, let’s analyze the expression sin(-\theta) / cos(-\theta). Here, we apply the even-odd identities for sine and cosine. We know that:

• sin(-\theta) = -sin(\theta)
• cos(-\theta) = cos(\theta)

Substituting these identities into the expression gives us:

sin(-\theta) / cos(-\theta) = -sin(\theta) / cos(\theta).

Recognizing that sin(\theta) / cos(\theta) is equivalent to tan(\theta), we can further simplify this to:

-tan(\theta).

Both methods of simplification lead to the same result, demonstrating the flexibility in using trigonometric identities. Mastery of these identities not only aids in simplifying expressions but also enhances problem-solving skills in trigonometry.

Use the even-odd identities to evaluate the expression.
$\cos\left(-\theta\right)-\cos\theta$

• A. 0
• B. $-\cos\theta$
• C. $2\cos\theta$
• D. $-2\cos\theta$

Answer: (A)

Use the even-odd identities to evaluate the expression.
$-\cot\left(\theta\right)\cdot\sin\left(-\theta\right)$

• A. $\tan\theta$
• B. $-\cos\theta$
• C. $\cos\theta$
• D. $\frac{\cos\theta}{\sin^2\theta}$

Answer: (C)

Select the expression with the same value as the given expression.
$\sec\left(-\frac{4\pi}{5}\right)$

• A. $\cos\left(\frac{4\pi}{5}\right)$
• B. $-\cos\left(\frac{4\pi}{5}\right)$
• C. $\sec\left(\frac{4\pi}{5}\right)$
• D. $-\sec\left(\frac{4\pi}{5}\right)$

Answer: (C)

Select the expression with the same value as the given expression.
$\sin\left(-38\degree\right)$

• A. $\sin38\degree$
• B. $-\sin38\degree$
• C. $-\sin\left(-38\degree\right)$
• D. $\frac{1}{-\sin38\degree}$

Answer: (B)

Pythagorean Identities

Understanding trigonometric expressions can be simplified significantly by utilizing Pythagorean identities. A fundamental identity states that the sine squared of an angle plus the cosine squared of that angle equals one. This can be expressed mathematically as:

$$\sin^2(\theta) + \cos^2(\theta) = 1$$

This identity is derived from the Pythagorean theorem, where in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. In the context of the unit circle, if we consider a triangle with a hypotenuse of 1, the lengths of the legs correspond to the sine and cosine of the angle, confirming that the identity holds true for any angle.

For example, for the angle \( \frac{\pi}{6} \), we know:

$$\sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \quad \text{and} \quad \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$$

Squaring these values gives:

$$\sin^2\left(\frac{\pi}{6}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$$

$$\cos^2\left(\frac{\pi}{6}\right) = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}$$

Adding these results confirms the identity:

$$\frac{1}{4} + \frac{3}{4} = 1$$

In addition to the primary identity, two other Pythagorean identities can be derived by manipulating the first. Dividing the first identity by \( \cos^2(\theta) \) yields:

$$\tan^2(\theta) + 1 = \sec^2(\theta)$$

Similarly, dividing by \( \sin^2(\theta) \) gives:

$$1 + \cot^2(\theta) = \csc^2(\theta)$$

These identities are useful for simplifying expressions involving squared trigonometric functions. For instance, if tasked with simplifying \( \sec^2(\theta) - \tan^2(\theta) \), one can apply the second Pythagorean identity:

$$\sec^2(\theta) - \tan^2(\theta) = 1$$

Another example involves the expression \( 1 - \cos(\theta) \times 1 + \cos(\theta) \). By recognizing this as a difference of squares, it can be rewritten as:

$$1 - \cos^2(\theta)$$

Using the first Pythagorean identity, we can express this as:

$$1 - \cos^2(\theta) = \sin^2(\theta)$$

Thus, the expression simplifies to \( \sin^2(\theta) \). Mastery of these identities allows for efficient simplification of various trigonometric expressions, enhancing problem-solving skills in trigonometry.

Example 2

In this problem, we are tasked with rewriting the expression \(\frac{1}{1 + \cos(\theta)}\) without a fraction by utilizing Pythagorean identities. Initially, it may not seem straightforward since there is no squared trigonometric function present. However, we can manipulate the expression to achieve our goal.

To begin, we multiply the denominator by \(1 - \cos(\theta)\). This gives us:

\(\frac{1 - \cos(\theta)}{(1 + \cos(\theta))(1 - \cos(\theta))}\)

Using the difference of squares in the denominator, we simplify it to:

\(1 - \cos^2(\theta)\)

Recognizing that \(1 - \cos^2(\theta)\) is equivalent to \(\sin^2(\theta)\) (from the Pythagorean identity), we can rewrite our expression as:

\(\frac{1 - \cos(\theta)}{\sin^2(\theta)}\)

To eliminate the fraction, we can express this as:

\(\frac{1}{\sin^2(\theta)}(1 - \cos(\theta))\)

Here, we note that \(\frac{1}{\sin(\theta)}\) is defined as \(\csc(\theta)\), thus \(\frac{1}{\sin^2(\theta)}\) becomes \(\csc^2(\theta)\). Therefore, we can rewrite the entire expression as:

\(\csc^2(\theta)(1 - \cos(\theta))\)

In conclusion, we have successfully transformed the original expression \(\frac{1}{1 + \cos(\theta)}\) into \(\csc^2(\theta)(1 - \cos(\theta))\), achieving our goal of eliminating the fraction.

Use the Pythagorean identities to rewrite the expression as a single term.
$\left(1+\csc\theta\right)\left(1-\csc\theta\right)$

• A. 1
• B. $-\csc^2\theta$
• C. $\cot^2\theta$
• D. $-\cot^2\theta$

Answer: (D)

Use the Pythagorean identities to rewrite the expression with no fraction.
$\frac{1}{1-\sec\theta}$

• A. $1+\sec\theta$
• B. $\frac{1}{\tan^2\theta}$
• C. $-\cot^2\theta\left(1+\sec\theta\right)$
• D. $-\tan^2\theta\left(1+\sec\theta\right)$

Answer: (C)

Simplifying Trig Expressions

Understanding how to simplify trigonometric expressions is essential in trigonometry, and it often involves applying various identities. When simplifying, it’s crucial to ensure that the expression meets three key criteria: all arguments must be positive, the expression should contain no fractions, and it should have as few trigonometric functions as possible.

For instance, consider the expression tan(-\theta) \cdot csc(\theta). To simplify this, we first address the negative argument. Using the odd identity for tangent, we know that tan(-\theta) = -tan(\theta). This transforms our expression to -tan(\theta) \cdot csc(\theta). Next, we can express tangent and cosecant in terms of sine and cosine: - \frac{\sin(\theta)}{\cos(\theta)} \cdot \frac{1}{\sin(\theta)}. Here, the sine terms cancel out, leaving us with -\frac{1}{\cos(\theta)}, which simplifies to \frac{\sin^2(\theta)}{1 + \cos(\theta)}. Initially, all arguments are positive, but the expression is a fraction. To eliminate the fraction, we can multiply both the numerator and denominator by 1 - \cos(\theta). This gives us \sin^2(\theta)(1 - \cos(\theta)) in the numerator and 1 - \cos^2(\theta) = \sin^2(\theta), we can replace the denominator, resulting in \frac{\sin^2(\theta)(1 - \cos(\theta))}{\sin^2(\theta)}. The sine terms cancel, leaving us with 1 - \cos(\theta), which is fully simplified.

These examples illustrate that while there are multiple strategies to simplify trigonometric expressions, the key is to remain flexible and apply the identities as needed. Mastery of these techniques will enhance your ability to tackle more complex trigonometric problems effectively.

Example 3

To simplify the expression \(\frac{\sin^2(\theta) - \tan^2(\theta)}{\sin(\theta) + \tan(\theta)}\), we start by recognizing that the numerator can be factored as a difference of squares. The difference of squares identity states that \(a^2 - b^2 = (a + b)(a - b)\). Here, we can identify \(a = \sin(\theta)\) and \(b = \tan(\theta)\), leading to:

\(\sin^2(\theta) - \tan^2(\theta) = (\sin(\theta) + \tan(\theta))(\sin(\theta) - \tan(\theta))\).

Substituting this back into our expression gives:

\(\frac{(\sin(\theta) + \tan(\theta))(\sin(\theta) - \tan(\theta))}{\sin(\theta) + \tan(\theta)}\).

Next, we can cancel the common term \((\sin(\theta) + \tan(\theta))\) from the numerator and the denominator, provided that \(\sin(\theta) + \tan(\theta) \neq 0\). This cancellation simplifies our expression to:

\(\sin(\theta) - \tan(\theta)\).

At this point, we have eliminated the fraction and reduced the expression to a simpler form. The final result, \(\sin(\theta) - \tan(\theta)\), contains two trigonometric functions, which is as simplified as we can get under the given conditions. Thus, the fully simplified expression is:

\(\sin(\theta) - \tan(\theta)\).

Example 4

To simplify the expression involving trigonometric functions, we start with the expression:

$$\frac{\cos(\theta) + \csc(\theta)}{\cos(\theta) + \sin(\theta) - \sec(\theta)}$$

Our goal is to simplify this expression by eliminating fractions and reducing the number of trigonometric functions. First, we identify the common denominator for the two fractions in the numerator and denominator. The common denominator is the product of the sine and cosine functions, specifically $$\sin(\theta) \cos(\theta)$$.

To combine the fractions, we multiply the first fraction by $$\frac{\sin(\theta)}{\sin(\theta)}$$ and the second fraction by $$\frac{\cos(\theta)}{\cos(\theta)}$$. This gives us:

$$\frac{\sin(\theta) \cos(\theta) + 1}{\sin(\theta) \cos(\theta) + \cos(\theta) \sin(\theta) - 1}$$

Next, we apply the identities for cosecant and secant, where $$\csc(\theta) = \frac{1}{\sin(\theta)}$$ and $$\sec(\theta) = \frac{1}{\cos(\theta)}$$. Substituting these into the expression, we have:

$$\frac{\sin(\theta) \cos(\theta) + 1}{\sin(\theta) \cos(\theta) + \cos(\theta) \sin(\theta) - 1}$$

Now, we simplify the numerator and denominator. The numerator simplifies to:

$$\sin(\theta) \cos(\theta) + 1$$

And the denominator simplifies to:

$$\sin(\theta) \cos(\theta) + \sin(\theta) \cos(\theta) - 1 = 2\sin(\theta) \cos(\theta) - 1$$

Next, we notice that the terms $$\sin(\theta) \cos(\theta)$$ in the numerator and denominator can be factored out. This leads us to:

$$\frac{2\sin(\theta) \cos(\theta)}{\sin(\theta) \cos(\theta)}$$

Upon canceling the common terms, we are left with:

$$2$$

This final result indicates that the original expression simplifies to a constant value of 2, demonstrating that we have successfully eliminated all fractions and reduced the number of trigonometric functions to none. Thus, the simplified expression is:

$$2$$

Simplify the expression.
$\tan^2\theta-\sec^2\theta+1$

• A. 0
• B. 1
• C. $\csc^2\theta+1$
• D. 2

Answer: (A)

Simplify the expression.
$\frac{\tan\left(-\theta\right)}{\sec\left(-\theta\right)}$

• A. $\sin\theta$
• B. $-\sin\theta$
• C. $-\cot\theta$
• D. 1

Answer: (B)

Simplify the expression.
$\left(\frac{\tan^2\theta}{\sin^2\theta}-1\right)\csc^2\left(\theta\right)\cos^2\left(-\theta\right)$

• A. $\cot^2\theta$
• B. $\tan\theta$
• C. 1
• D. – 1

Answer: (C)

Verifying Trig Equations as Identities

Verifying trigonometric identities involves demonstrating that two sides of an equation are equal by simplifying one or both sides. This process relies heavily on recognizing and applying trigonometric identities, particularly Pythagorean identities, which are fundamental in transforming expressions.

For instance, consider the identity:

\[\frac{\sin \theta \cdot \cos \theta}{1 - \cos^2 \theta} = \frac{1}{\tan \theta}\]

To verify this, start with the more complex side, which is the left side in this case. The denominator, \(1 - \cos^2 \theta\), can be simplified using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1\). Rearranging gives \(1 - \cos^2 \theta = \sin^2 \theta\). Thus, the left side simplifies to:

\[\frac{\sin \theta \cdot \cos \theta}{\sin^2 \theta} = \frac{\cos \theta}{\sin \theta} = \cot \theta\]

Recognizing that \(\cot \theta = \frac{1}{\tan \theta}\), we conclude that both sides are equal, confirming the identity.

In another example, consider the identity:

\[\frac{\sec^2 \theta - \tan^2 \theta}{\cos(-\theta) + 1} = \frac{1 - \cos \theta}{\sin^2 \theta}\]

Starting with the left side, we apply the Pythagorean identity again, where \(\sec^2 \theta - \tan^2 \theta = 1\). This simplifies the numerator to 1. The denominator, \(\cos(-\theta) + 1\), simplifies to \(\cos \theta + 1\) due to the even property of cosine. Thus, the left side becomes:

\[\frac{1}{\cos \theta + 1}\]

Next, simplifying the right side involves multiplying the numerator and denominator by \(1 + \cos \theta\) to utilize the difference of squares:

\[\frac{(1 - \cos \theta)(1 + \cos \theta)}{\sin^2 \theta(1 + \cos \theta)} = \frac{1 - \cos^2 \theta}{\sin^2 \theta} = \frac{\sin^2 \theta}{\sin^2 \theta(1 + \cos \theta)} = \frac{1}{1 + \cos \theta}\]

Both sides are now equal, confirming the identity.

When verifying identities, it is crucial to remain vigilant for opportunities to apply known identities and to simplify expressions methodically. If you encounter difficulties, revisiting the problem from a fresh perspective can often reveal new insights. With practice, your proficiency in verifying trigonometric identities will improve significantly.

Example 5

In this problem, we are tasked with verifying the identity by manipulating one side of the equation. The equation presented is:

\[\frac{1 - \sin(\theta)}{\cos(\theta)} - \frac{\cos(\theta)}{1 + \sin(\theta)} = 0\]

To begin, we focus on the left side of the equation since the right side is already simplified to 0. The first step involves combining the two fractions by finding a common denominator. The denominators in this case are \(\cos(\theta)\) and \(1 + \sin(\theta)\). The common denominator will be the product of these two, which is \(\cos(\theta)(1 + \sin(\theta))\).

Next, we adjust each fraction to have this common denominator. For the first fraction, we multiply both the numerator and denominator by \(1 + \sin(\theta)\), resulting in:

\[\frac{(1 - \sin(\theta))(1 + \sin(\theta))}{\cos(\theta)(1 + \sin(\theta))} = \frac{1 - \sin^2(\theta)}{\cos(\theta)(1 + \sin(\theta))}\]

For the second fraction, we multiply both the numerator and denominator by \(\cos(\theta)\), yielding:

\[\frac{-\cos^2(\theta)}{\cos(\theta)(1 + \sin(\theta))}\]

Now, we can combine these two fractions over the common denominator:

\[\frac{1 - \sin^2(\theta) - \cos^2(\theta)}{\cos(\theta)(1 + \sin(\theta))}\]

At this point, we can apply the Pythagorean identity, which states that \(\sin^2(\theta) + \cos^2(\theta) = 1\). Rearranging this gives us \(1 - \sin^2(\theta) = \cos^2(\theta)\). Substituting this into our expression, we have:

\[\frac{\cos^2(\theta) - \cos^2(\theta)}{\cos(\theta)(1 + \sin(\theta))}\]

In the numerator, \(\cos^2(\theta) - \cos^2(\theta)\) simplifies to 0. Thus, we are left with:

\[\frac{0}{\cos(\theta)(1 + \sin(\theta))} = 0\]

Since the left side simplifies to 0, and the right side of the original equation is also 0, we have successfully verified the identity:

\[0 = 0\]

This confirms that the identity holds true. If there are any questions or further clarifications needed, feel free to ask!

Example 6

To verify the trigonometric identity given by the equation:

\[\sec(\theta) \cdot (1 - \sin^2(\theta)) = \frac{1 + \tan^2(\theta) \cdot \cot^2(\theta)}{\csc^2(\theta) \cdot \sec(\theta)}\]

we start by simplifying the more complex side, which is the right side of the equation. The expression can be rewritten using known trigonometric identities. Notably, we recognize that:

\[1 + \tan^2(\theta) = \sec^2(\theta)\]

Thus, we can substitute this into the equation, leading to:

\[\frac{\sec^2(\theta) \cdot \cot^2(\theta)}{\csc^2(\theta) \cdot \sec(\theta)}\]

Next, we can simplify this expression by canceling one \(\sec(\theta)\) from the numerator and denominator:

\[\frac{\sec(\theta) \cdot \cot^2(\theta)}{\csc^2(\theta)}\]

Now, we express everything in terms of sine and cosine. Recall that:

• \(\sec(\theta) = \frac{1}{\cos(\theta)}\)
• \(\cot^2(\theta) = \frac{\cos^2(\theta)}{\sin^2(\theta)}\)
• \(\csc^2(\theta) = \frac{1}{\sin^2(\theta)}\)

Substituting these identities into our expression gives:

\[\frac{\frac{1}{\cos(\theta)} \cdot \frac{\cos^2(\theta)}{\sin^2(\theta)}}{\frac{1}{\sin^2(\theta)}}\]

When we simplify this, we multiply by the reciprocal of the denominator:

\[\frac{\cos^2(\theta)}{\sin^2(\theta)} \cdot \sin^2(\theta) = \cos(\theta)\]

Now, we turn our attention to the left side of the original identity:

\[\sec(\theta) \cdot (1 - \sin^2(\theta))\]

Using the Pythagorean identity \(1 - \sin^2(\theta) = \cos^2(\theta)\), we can rewrite the left side as:

\[\sec(\theta) \cdot \cos^2(\theta)\]

Substituting \(\sec(\theta)\) gives us:

\[\frac{1}{\cos(\theta)} \cdot \cos^2(\theta) = \cos(\theta)\]

Both sides of the equation simplify to \(\cos(\theta)\), confirming that the identity is verified:

\[\cos(\theta) = \cos(\theta)\]

Identify the most helpful first step in verifying the identity.
$\left(\frac{\tan^2\theta}{\sin^2\theta}-1\right)=\sec^2\theta\sin^2\left(-\theta\right)$

• A. Add the terms on the left side using a common denominator.
• B. Rewrite left side of equation in terms of sine and cosine.
• C. Use even-odd identity to eliminate negative argument on right side of equation.
• D. Rewrite right side of equation in terms of sine and cosine.

Answer: (B)

Identify the most helpful first step in verifying the identity.
$\sec^3\theta=\sec\theta+\frac{\tan^2\theta}{\cos\theta}$

• A. Rewrite left side of equation in terms of sine and cosine.
• B. Subtract $\sec\theta$ from both sides.
• C. Use reciprocal identity to rewrite $\sec\theta$ on right side of equation.
• D. Rewrite $\tan^2\theta$ in terms of sine and cosine.

Answer: (C)

Here’s what students ask on this topic:

What are the even and odd trigonometric identities?

Even and odd trigonometric identities help simplify expressions involving trigonometric functions. The cosine function is even, meaning $\cos (-\theta )=\cos \left(\theta \right)$. The sine function is odd, so $\sin (-\theta )=-\sin \left(\theta \right)$. Similarly, secant is even, and cosecant, tangent, and cotangent are odd. These identities are useful when dealing with negative arguments in trigonometric functions.

How do you use Pythagorean identities to simplify trigonometric expressions?

Pythagorean identities are essential for simplifying trigonometric expressions. The primary identity is ${\sin }^2\left(\theta \right)+{\cos }^2\left(\theta \right)=1$. By dividing this identity by ${\cos }^2\left(\theta \right)$ or ${\sin }^2\left(\theta \right)$, you get the other two identities: ${\tan }^2\left(\theta \right)+1={\sec }^2\left(\theta \right)$ and $1+{\cot }^2\left(\theta \right)={\csc }^2\left(\theta \right)$. These identities help in rewriting and simplifying expressions involving squared trigonometric functions.

How do you verify a trigonometric identity?

Verifying a trigonometric identity involves showing that both sides of an equation are equal. Start by simplifying the more complicated side using known identities like Pythagorean, even-odd, and reciprocal identities. For example, to verify $\sin \left(\theta \right)\cos \left(\theta \right)/(1-{\cos }^2(\theta \left)\right)=1/\tan \left(\theta \right)$, simplify the left side using the identity $1-{\cos }^2\left(\theta \right)={\sin }^2\left(\theta \right)$. This reduces to $\cos \left(\theta \right)/\sin \left(\theta \right)=1/\tan \left(\theta \right)$, proving the identity.

What are some strategies for simplifying trigonometric expressions?

Simplifying trigonometric expressions involves several strategies: 1) Constantly scan for identities like Pythagorean, even-odd, and reciprocal identities. 2) Break down complex expressions into sine and cosine. 3) Use algebraic techniques like factoring and combining fractions. 4) If you encounter $1+\cos \left(\theta \right)$ or $1-\cos \left(\theta \right)$, multiply by the conjugate. For example, to simplify ${\sin }^2(\theta )/(1+\cos \left(\theta \right))$, multiply by $1-\cos \left(\theta \right)$ to get ${\sin }^2(\theta )/{\sin }^2(\theta )=1$.

How do you determine if a trigonometric function is even or odd?

To determine if a trigonometric function is even or odd, analyze its symmetry. A function is even if $f(-x)=f\left(x\right)$ and odd if $f(-x)=-f\left(x\right)$. For trigonometric functions, cosine and secant are even because $\cos (-\theta )=\cos \left(\theta \right)$ and $\sec (-\theta )=\sec \left(\theta \right)$. Sine, tangent, cosecant, and cotangent are odd because $\sin (-\theta )=-\sin \left(\theta \right)$ and similarly for the others. This property helps in simplifying expressions with negative arguments.