# Introduction to Trigonometric Identities - Video Tutorials & Practice Problems

## Even and Odd Identities

## Example 1

Use the even-odd identities to evaluate the expression.

$\cos\left(-\theta\right)-\cos\theta$

0

$-\cos\theta$

$2\cos\theta$

$-2\cos\theta$

Use the even-odd identities to evaluate the expression.

$-\cot\left(\theta\right)\cdot\sin\left(-\theta\right)$

$\tan\theta$

$-\cos\theta$

$\cos\theta$

$\frac{\cos\theta}{\sin^2\theta}$

Select the expression with the same value as the given expression.

$\sec\left(-\frac{4\pi}{5}\right)$

$\cos\left(\frac{4\pi}{5}\right)$

$-\cos\left(\frac{4\pi}{5}\right)$

$\sec\left(\frac{4\pi}{5}\right)$

$-\sec\left(\frac{4\pi}{5}\right)$

Select the expression with the same value as the given expression.

$\sin\left(-38\degree\right)$

$\sin38\degree$

$-\sin38\degree$

$-\sin\left(-38\degree\right)$

$\frac{1}{-\sin38\degree}$

## Pythagorean Identities

## Example 2

Use the Pythagorean identities to rewrite the expression as a single term.

$\left(1+\csc\theta\right)\left(1-\csc\theta\right)$

1

$-\csc^2\theta$

$\cot^2\theta$

$-\cot^2\theta$

Use the Pythagorean identities to rewrite the expression with no fraction.

$\frac{1}{1-\sec\theta}$

$1+\sec\theta$

$\frac{1}{\tan^2\theta}$

$-\cot^2\theta\left(1+\sec\theta\right)$

$-\tan^2\theta\left(1+\sec\theta\right)$

## Simplifying Trig Expressions

## Example 3

## Example 4

Simplify the expression.

$\tan^2\theta-\sec^2\theta+1$

0

1

$\csc^2\theta+1$

2

Simplify the expression.

$\frac{\tan\left(-\theta\right)}{\sec\left(-\theta\right)}$

$\sin\theta$

$-\sin\theta$

$-\cot\theta$

1

Simplify the expression.

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

$\cot^2\theta$

$\tan\theta$

1

– 1

## Verifying Trig Equations as Identities

## Example 5

## Example 6

Identify the most helpful first step in verifying the identity.

$\left(\frac{\tan^2\theta}{\sin^2\theta}-1\right)=\sec^2\theta\sin^2\left(-\theta\right)$

Add the terms on the left side using a common denominator.

Rewrite left side of equation in terms of sine and cosine.

Use even-odd identity to eliminate negative argument on right side of equation.

Rewrite right side of equation in terms of sine and cosine.

Identify the most helpful first step in verifying the identity.

$\sec^3\theta=\sec\theta+\frac{\tan^2\theta}{\cos\theta}$

Rewrite left side of equation in terms of sine and cosine.

Subtract $\sec\theta$ from both sides.

Use reciprocal identity to rewrite $\sec\theta$ on right side of equation.

Rewrite $\tan^2\theta$ in terms of sine and cosine.

## Do you want more practice?

- In Exercises 1–60, verify each identity. sin x sec x = tan x
- In Exercises 1–60, verify each identity. tan (-x) cos x = -sin x
- In Exercises 1–60, verify each identity. tan x csc x cos x = 1
- Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients ap...
- Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients ap...
- Perform each indicated operation and simplify the result so that there are no quotients.sec x/csc x + csc x/se...
- Work each problem.Given tan x = -5⁄4, where π/2< x < π, use the trigonometric identities to find cot x, ...
- Perform each indicated operation and simplify the result so that there are no quotients.cos β(sec β + csc β)
- Work each problem.Find the exact values of sin x, cos x, and tan x, for x = π/12 , usinga. difference identiti...
- Perform each indicated operation and simplify the result so that there are no quotients.cos x/sec x + sin x/cs...
- For each expression in Column I, use an identity to choose an expression from Column II with the same value. C...
- Perform each indicated operation and simplify the result so that there are no quotients.(tan x + cot x)²
- For each expression in Column I, use an identity to choose an expression from Column II with the same value. C...
- For each expression in Column I, choose the expression from Column II that completes an identity.2. csc x = __...
- Perform each indicated operation and simplify the result so that there are no quotients.(1 + tan θ)² - 2 tan θ
- For each expression in Column I, use an identity to choose an expression from Column II with the same value. C...
- Perform each indicated operation and simplify the result so that there are no quotients.1/( sin α - 1) - 1/(si...
- Factor each trigonometric expression.sec² θ - 1
- Factor each trigonometric expression.(tan x + cot x)² - (tan x - cot x)²
- Factor each trigonometric expression.4 tan² β + tan β - 3
- Factor each trigonometric expression.cot⁴ x + 3 cot² x + 2
- Factor each trigonometric expression.sin³ α + cos³ α
- Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identit...
- Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identit...
- Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identit...
- For each expression in Column I, choose the expression from Column II that completes an identity. One or both ...
- For each expression in Column I, choose the expression from Column II that completes an identity.4. cot x = __...
- For each expression in Column I, choose the expression from Column II that completes an identity. One or both ...
- For each expression in Column I, choose the expression from Column II that completes an identity. One or both ...
- Concept Check Suppose that sec θ = (x+4)/x.Find an expression in x for tan θ.
- Verify that each equation is an identity.tan α/sec α = sin α
- Perform each transformation. See Example 2.Write cot x in terms of sin x.
- Verify that each equation is an identity.(tan² α + 1)/ sec α = sec α
- Perform each transformation. See Example 2.Write cot x in terms of csc x.
- Verify that each equation is an identity.sin² β (1 + cot² β) = 1
- Verify that each equation is an identity.2 cos³ x - cos x = (cos² x - sin² x)/sec x
- Perform each transformation. See Example 2.Write sec x in terms of sin x.
- Verify that each equation is an identity.sin² α + tan² α + cos² α = sec² α
- Verify that each equation is an identity.(sin 2x)/(sin x) = 2/sec x
- Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appea...
- Verify that each equation is an identity.(sin² θ)/cos θ = sec θ - cos θ
- Verify that each equation is an identity.(2 tan B)/(sin 2B) = sec² B
- Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appea...
- Verify that each equation is an identity.sec⁴ x - sec² x = tan⁴ x + tan² x
- Verify that each equation is an identity.(2 cot x)/(tan 2x) = csc² x - 2
- Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appea...
- Verify that each equation is an identity.(sec α - tan α)² = (1 - sin α)/(1 + sin α)
- Verify that each equation is an identity.csc A sin 2A - sec A = cos 2A sec A
- For each expression in Column I, choose the expression from Column II that completes an identity.6. sec² x = _...
- Verify that each equation is an identity.[(sec θ - tan θ)² + 1]/(sec θ csc θ - tan θ csc θ) = 2 tan θ
- Verify that each equation is an identity.2 cos² θ - 1 = (1 - tan² θ)/(1 + tan² θ)
- Verify that each equation is an identity.1/(sec α - tan α) = sec α + tan α
- Verify that each equation is an identity.sec² α - 1 = (sec 2α - 1)/(sec 2α + 1)
- Verify that each equation is an identity.(csc θ + cot θ)/(tan θ + sin θ) = cot θ csc θ
- Verify that each equation is an identity.sin³ θ = sin θ - cos² θ sin θ
- Verify that each equation is an identity.sin² θ (1 + cot² θ) - 1 = 0
- Verify that each equation is an identity.2 cos² (x/2) tan x = tan x+ sin x
- Verify that each equation is an identity.(sin⁴ α - cos⁴ α )/(sin² α - cos² α) = 1
- Verify that each equation is an identity.(1/2)cot (x/2) - (1/2) tan (x/2) = cot x
- Verify that each equation is an identity.(cot² t - 1)/(1 + cot² t) = 1 - 2 sin² t
- Verify that each equation is an identity.(sin 3t + sin 2t)/(sin 3t - sin 2t ) = tan (5t/2)/(tan (t/2))
- Verify that each equation is an identity.tan² α sin² α = tan² α + cos² α - 1
- Verify that each equation is an identity.sin θ/(1 - cos θ) - sin θ cos θ/( 1 + cos θ) = csc θ (1 + cos² θ) ...
- Verify that each equation is an identity.(1 + sin θ)/(1 - sin θ) - (1 - sin θ)/( 1 + sin θ) = 4 tan θ sec θ
- Verify that each equation is an identity.sin θ + cos θ = sin θ/(1 - cot θ) + cos θ/(1 - tan θ)
- Let csc x = -3. Find all possible values of (sin x + cos x)/sec x.
- Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients ap...
- Verify that each equation is an identity.(1 + sin x + cos x)² = 2(1 + sin x) (1 + cos x)
- Verify that each equation is an identity.(sec α + csc α) (cos α - sin α) = cot α - tan α
- Verify that each equation is an identity.(1 - cos θ)/(1 + cos θ) = 2 csc² θ - 2 csc θ cot θ - 1
- Verify that each equation is an identity.sin² x(1 + cot x) + cos² x(1 - tan x) + cot² x = csc² x
- Verify that each equation is an identity.sin³ θ + cos³ θ = (cos θ + sin θ) (1 - cos θ sin θ)
- In Exercises 1–60, verify each identity. csc θ - sin θ = cot θ cos θ
- In Exercises 1–60, verify each identity. cos θ sec θ ----------------- = tan θ cot θ
- In Exercises 1–60, verify each identity. cos² θ (1 + tan² θ) = 1
- In Exercises 1–60, verify each identity. cot² t ------------ = csc t - sin t csc t