Trigonometric Identities: Videos & Practice Problems

To simplify trigonometric expressions, ensure all arguments are positive, eliminate fractions, and minimize the number of trig functions. Use identities like the Pythagorean identity, which states that ${\sin }^2+{\cos }^2=1$, to rewrite complex expressions. When verifying identities, simplify the more complex side first, aiming to show both sides are equal. This approach allows for solving linear trig equations effectively by reducing them to a single trig function.

Simplifying Trig Expressions

Understanding how to simplify trigonometric expressions is essential in trigonometry. To fully simplify a trigonometric expression, there are three key criteria to consider: all arguments must be positive, the expression should contain no fractions, and it should have as few trigonometric functions as possible.

For example, consider the expression tan(-θ) * csc(θ). To simplify this, we first apply the even-odd identities. The tangent of a negative angle can be rewritten as -tan(θ), transforming our expression to -tan(θ) * csc(θ). Now, both arguments are positive.

Next, we aim to reduce the number of trigonometric functions. The tangent can be expressed as sin(θ)/cos(θ) and cosecant as 1/sin(θ). Substituting these identities gives us:

$$ -\frac{\sin(θ)}{\cos(θ)} * \frac{1}{\sin(θ)} = -\frac{1}{\cos(θ)} $$

Recognizing that 1/cos(θ) is the secant function, we simplify further to -sec(θ). At this point, we have met all criteria for simplification.

In another example, consider the expression sin²(θ)/(1 + cos(θ)). Here, we notice that the expression is a single fraction. To eliminate the fraction, we can multiply both the numerator and denominator by 1 - cos(θ), leading to:

$$ \frac{\sin²(θ)(1 - cos(θ))}{(1 + cos(θ))(1 - cos(θ))} $$

The denominator simplifies to a difference of squares:

$$ 1 - cos²(θ) $$

Using the Pythagorean identity sin²(θ) + cos²(θ) = 1, we can replace 1 - cos²(θ) with sin²(θ). This gives us:

$$ \frac{\sin²(θ)(1 - cos(θ))}{sin²(θ)} $$

After canceling sin²(θ), we are left with 1 - cos(θ), which meets all criteria for simplification.

In summary, simplifying trigonometric expressions involves recognizing identities, reducing fractions, and minimizing the number of trigonometric functions. There are often multiple valid approaches to reach the simplified form, so exploring different methods can enhance understanding and flexibility in problem-solving.

Simplifying Trig Expressions Example 1

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 formula 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 simplify the expression by canceling the common term \((\sin(\theta) + \tan(\theta))\) in the numerator and denominator, provided that \(\sin(\theta) + \tan(\theta) \neq 0\). This cancellation results in:

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

At this point, we have eliminated the fraction and reduced the expression to a simpler form. The final simplified expression is:

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

This expression contains two trigonometric functions, which is the minimum number possible given the nature of the subtraction. Thus, we have fully simplified the original expression.

Simplifying Trig Expressions Example 2

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

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

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

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

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

Next, we apply trigonometric identities to replace \(\csc(\theta)\) with \(\frac{1}{\sin(\theta)}\) and \(\sec(\theta)\) with \(\frac{1}{\cos(\theta)}\). This leads to:

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

Now, we simplify the numerator and denominator. In the numerator, we have \(\sin(\theta) \cdot \cos(\theta) + 1\). In the denominator, we can use the identity \(\sin^2(\theta) + \cos^2(\theta) = 1\) to rewrite it as:

\[\sin(\theta) \cdot \cos(\theta) + 1 - 1 = \sin(\theta) \cdot \cos(\theta)\]

Thus, the expression simplifies to:

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

Now, we can separate the terms in the numerator:

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

This simplifies to:

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

Finally, we recognize that \(\frac{1}{\sin(\theta) \cdot \cos(\theta)}\) can be expressed as \(2\) when considering the double angle identity for sine, leading us to the final simplified result:

\[2\]

Thus, the original expression simplifies to \(2\), confirming that we have achieved a fully simplified form with no fractions and minimal trigonometric functions.

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 in this case is the left side. Recognizing that \(1 - \cos^2 \theta\) can be replaced using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1\), we can rewrite the denominator as \(\sin^2 \theta\). This transforms the left side into:

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

Since \(\cot \theta\) is equal to \(\frac{1}{\tan \theta}\), we conclude that both sides are equal, thus verifying 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 can apply the Pythagorean identity again, where \(\sec^2 \theta - \tan^2 \theta = 1\). This simplifies the numerator to 1. The denominator can be simplified using the even-odd identity, recognizing that \(\cos(-\theta) = \cos \theta\), leading to:

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

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

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

Both sides are now equal, confirming the identity.

When tackling these problems, if you find yourself stuck, it can be beneficial to start over or approach the problem from a different angle. With practice, your ability to recognize and apply these identities will improve, making the verification process more intuitive.

Verifying Trig Equations as Identities Example 3

To verify the identity given by the equation:

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

we will focus on simplifying the left side. The first step involves combining the two fractions by finding a common denominator. The denominators are \(\cos(\theta)\) and \(1 + \sin(\theta)\), so the common denominator will be \(\cos(\theta)(1 + \sin(\theta))\).

To achieve this, we multiply the first fraction by \((1 + \sin(\theta))\) in both the numerator and denominator:

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

Expanding the numerator using the difference of squares gives:

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

Next, we handle the second fraction by multiplying both the numerator and denominator by \(\cos(\theta)\):

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

Now, we can combine the two fractions:

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

Recognizing the Pythagorean identity, where \(1 - \sin^2(\theta) = \cos^2(\theta)\), we substitute this into the numerator:

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

This simplifies to:

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

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

\[0 = 0\]

This confirms that the left side equals the right side, completing the verification process.

Verifying Trig Equations as Identities Example 4

To verify the trigonometric identity involving secant, tangent, cotangent, cosecant, and sine, we start with the more complex side of the equation. The identity states that the secant of theta multiplied by (1 minus the sine squared of theta) is equal to the expression on the right side: (1 plus the tangent squared of theta) multiplied by the cotangent squared of theta, divided by the product of cosecant squared of theta and secant of theta.

We begin by simplifying the right side. Notably, we can use the Pythagorean identity, which states that \(1 + \tan^2(\theta) = \sec^2(\theta)\). By substituting this into our expression, we rewrite the numerator as \(\sec^2(\theta) \cdot \cot^2(\theta)\) over \(\csc^2(\theta) \cdot \sec(\theta)\). This allows us to cancel one secant from the numerator and denominator, simplifying our expression to \(\sec(\theta) \cdot \cot^2(\theta) / \csc^2(\theta)\).

Next, we express everything in terms of sine and cosine. We know that \(\sec(\theta) = \frac{1}{\cos(\theta)}\) and \(\cot^2(\theta) = \frac{\cos^2(\theta)}{\sin^2(\theta)}\). The denominator, \(\csc^2(\theta)\), can be rewritten as \(\frac{1}{\sin^2(\theta)}\). Substituting these identities into our expression, we have:

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

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

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

Now, we turn our attention to the left side of the original identity. The expression \(\sec(\theta) \cdot (1 - \sin^2(\theta))\) can be simplified using the Pythagorean identity \(1 - \sin^2(\theta) = \cos^2(\theta)\). Thus, we rewrite the left side as:

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

Substituting \(\sec(\theta) = \frac{1}{\cos(\theta)}\) gives us:

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

Both sides of the identity simplify to \(\cos(\theta)\), confirming that the original identity is verified. Therefore, we conclude that:

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

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)

Solve Trig Equations Using Identity Substitutions

When solving linear trigonometric equations, the goal is to find an angle \( \theta \) that satisfies the equation. As equations become more complex, involving multiple trigonometric functions, the process requires the application of trigonometric identities to simplify the expressions. For instance, consider the equation:

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

To solve this, we can utilize the Pythagorean identity, which states that \( \sec^2(\theta) - 1 = \tan^2(\theta) \). By substituting this identity into the equation, we rewrite it as:

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

Next, we can simplify the left side by canceling one \( \tan(\theta) \), resulting in:

\[\tan(\theta) = 1\]

From here, we can find the angles \( \theta \) on the unit circle where the tangent is equal to 1, which occurs at \( \theta = \frac{\pi}{4} + n\pi \), where \( n \) is any integer, to account for all possible solutions.

In another example, consider the equation:

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

Here, we can apply the double angle identity for sine, which gives us:

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

Next, we can simplify by recognizing that \( \cos(-\theta) = \cos(\theta) \) due to the even nature of the cosine function. This leads to:

\[2\sin(\theta) = 1\]

To isolate \( \sin(\theta) \), we divide both sides by 2:

\[\sin(\theta) = \frac{1}{2}\]

The angles \( \theta \) that satisfy this equation on the unit circle are \( \theta = \frac{\pi}{6} \) and \( \theta = \frac{5\pi}{6} \). To find all solutions, we add \( 2\pi n \) to each angle, resulting in:

\[\theta = \frac{\pi}{6} + 2\pi n \quad \text{and} \quad \theta = \frac{5\pi}{6} + 2\pi n\]

By employing trigonometric identities and simplification techniques, we can effectively solve more complex trigonometric equations, ensuring we account for all possible solutions.

Find all solutions to the equation.
$\left(\cos\theta+\sin\theta\right)\left(\cos\theta-\sin\theta\right)=-\frac12$

• A. $\theta=\frac{5\pi}{12}+2\pi n,\frac{7\pi}{12}+2\pi n$
• B. $\theta=\frac{2\pi}{3}+2\pi n,\frac{4\pi}{3}+2\pi n$
• C. $\theta=\frac{\pi}{3}+2\pi n,\frac{2\pi}{3}+2\pi n$
• D. $\theta=\frac{\pi}{3}+\pi n,\frac{2\pi}{3}+\pi n$

Answer: (D)

Find all solutions to the equation where 0 ≤ $\theta$ ≤ $2\pi$.
$\sin\theta\cos\left(2\theta\right)-\sin\left(2\theta\right)\cos\theta=\frac{\sqrt2}{2}$

• A. $\theta=\frac{5\pi}{4},\frac{7\pi}{4}$
• B. $\theta=\frac{\pi}{4},\frac{3\pi}{4}$
• C. $\theta=\frac{5\pi}{4}+2\pi n,\frac{7\pi}{4}+2\pi n$
• D. $\theta=\frac{3\pi}{4},\frac{7\pi}{4}$

Answer: (A)

Here’s what students ask on this topic:

What are the Pythagorean trigonometric identities?

The Pythagorean trigonometric identities are fundamental relationships between the sine, cosine, and tangent functions. They are derived from the Pythagorean theorem and are as follows:

${\sin }^2\theta +{\cos }^2\theta =\; 1$

${\tan }^2\theta +\; 1\; ={\sec }^2\theta$

$1\; +{\cot }^2\theta ={\csc }^2\theta$

These identities are useful for simplifying trigonometric expressions and solving trigonometric equations.

How do you simplify trigonometric expressions?

To simplify trigonometric expressions, follow these steps:

1. Ensure all arguments are positive by using even-odd identities.

2. Eliminate fractions by multiplying by the conjugate or using identities.

3. Minimize the number of trigonometric functions by expressing everything in terms of sine and cosine.

For example, to simplify $\tan -\theta \times \csc \theta$, use the identities $\tan -\theta =-\tan \theta$ and $\csc \theta =\frac{1}{\sin }$. This simplifies to $-\frac{\sin }{\theta }\times \frac{1}{\sin }=-\frac{1}{\cos }=-\sec \theta$.

How do you verify trigonometric identities?

To verify trigonometric identities, follow these steps:

1. Simplify the more complex side of the equation first.

2. Use trigonometric identities to rewrite the expression.

3. Aim to show both sides of the equation are equal.

For example, to verify $\frac{\sin }{\theta }=\tan \theta$, start with the left side and use the identity $\tan \theta =\frac{\sin }{\theta }$. This shows that both sides are equal, verifying the identity.

How do you solve trigonometric equations using identities?

To solve trigonometric equations using identities, follow these steps:

1. Use identities to rewrite the equation in terms of a single trigonometric function.

2. Solve the resulting linear trigonometric equation.

For example, to solve $\frac{\sec }{{\theta }^2}=1$, use the identity $\sec {\theta }^2-1=\tan {\theta }^2$ to rewrite the equation as $\frac{\tan }{{\theta }^2}=1$. This simplifies to $\tan \theta =1$. Find the angles $\theta =\frac{\pi }{4}+\mathrm{n}\pi$ for all solutions.

What are even and odd trigonometric identities?

Even and odd trigonometric identities help simplify expressions with negative arguments. The even identities are:

$\cos -\theta =\cos \theta$

$\sec -\theta =\sec \theta$

The odd identities are:

$\sin -\theta =-\sin \theta$

$\tan -\theta =-\tan \theta$

$\csc -\theta =-\csc \theta$

$\cot -\theta =-\cot \theta$

These identities are useful for simplifying trigonometric expressions and solving equations.