Trigonometry Practice

Trigonometric Identities Practice

Practice recognizing, rewriting, simplifying, evaluating, and verifying the major families of trigonometric identities.

Start Practicing Review Identities Identity Reference

Try each problem first, then click Show solution to check the identity you chose, your algebra, and your final result.

Keep These Relationships Nearby

Trigonometric Identities Essentials

Quotient

\[ \tan\theta = \frac{\sin\theta}{\cos\theta} \]

Pythagorean

\[ \sin^2\theta+\cos^2\theta=1 \]

Double Angle

\[ \sin(2\theta) = 2\sin\theta\cos\theta \]
Download Practice PDF Identity Reference Review Identities

Level 1

Recognize the Identity

Start by matching familiar forms with the correct identity family.

Problem 1

Identify the Family

Which identity family contains \[ 1+\tan^2\theta=\sec^2\theta? \]

Reciprocal
Quotient
Pythagorean

Step-by-Step Solution

The equation relates a squared tangent expression, a \(1\), and a squared secant expression.

\[ 1+\tan^2\theta = \sec^2\theta \]
Answer: Pythagorean identity

Problem 2

Match Equivalent Forms

Which expression is equivalent to \(\sec\theta\)?

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

Step-by-Step Solution

Secant is the reciprocal of cosine.

\[ \sec\theta = \frac{1}{\cos\theta} \]
Answer: \[ \boxed{ \frac{1}{\cos\theta} } \]

Problem 3

Recognize a Rearrangement

Which expression is equivalent to \(1-\cos^2\theta\)?

\[ \sin^2\theta \]
\[ \tan^2\theta \]
\[ \sec^2\theta \]

Step-by-Step Solution

Start with the fundamental Pythagorean identity.

\[ \sin^2\theta + \cos^2\theta = 1 \] \[ \sin^2\theta = 1-\cos^2\theta \]
Answer: \[ \boxed{\sin^2\theta} \]

Level 2

Rewrite and Simplify

Use reciprocal, quotient, and Pythagorean identities to simplify expressions.

Problem 4

Simplify

Simplify: \[ \frac{\tan\theta}{\sec\theta} \]

Step-by-Step Solution

Rewrite tangent and secant using sine and cosine.

\[ \frac{\tan\theta}{\sec\theta} = \frac{ \frac{\sin\theta}{\cos\theta} }{ \frac{1}{\cos\theta} } \] \[ = \frac{\sin\theta}{\cos\theta} \cdot \cos\theta = \sin\theta \]
Answer: \[ \boxed{\sin\theta} \]

Problem 5

Simplify

Simplify: \[ \frac{ 1-\cos^2\theta }{ \sin\theta } \]

Step-by-Step Solution

Use \(1-\cos^2\theta = \sin^2\theta\).

\[ \frac{ 1-\cos^2\theta }{ \sin\theta } = \frac{ \sin^2\theta }{ \sin\theta } = \sin\theta \]
Answer: \[ \boxed{\sin\theta} \]

Problem 6

Simplify

Simplify: \[ \frac{ \sec^2\theta-1 }{ \tan\theta } \]

Step-by-Step Solution

Use \(\sec^2\theta-1 = \tan^2\theta\).

\[ \frac{ \sec^2\theta-1 }{ \tan\theta } = \frac{ \tan^2\theta }{ \tan\theta } = \tan\theta \]
Answer: \[ \boxed{\tan\theta} \]

Level 3

Sum and Difference Identities

Use sum and difference formulas to find exact values and rewrite expressions.

Problem 7

Exact Value

Find the exact value of \[ \sin75^\circ. \]

Step-by-Step Solution

Write \(75^\circ = 45^\circ+30^\circ\) and use the sine sum identity.

\[ \sin75^\circ = \sin45^\circ\cos30^\circ + \cos45^\circ\sin30^\circ \] \[ = \frac{\sqrt2}{2} \frac{\sqrt3}{2} + \frac{\sqrt2}{2} \frac12 = \frac{ \sqrt6+\sqrt2 }{ 4 } \]
Answer: \[ \boxed{ \frac{ \sqrt6+\sqrt2 }{ 4 } } \]

Problem 8

Exact Value

Find the exact value of \[ \cos15^\circ \] using a difference identity.

Step-by-Step Solution

Write \(15^\circ = 45^\circ-30^\circ\).

\[ \cos15^\circ = \cos45^\circ\cos30^\circ + \sin45^\circ\sin30^\circ \] \[ = \frac{\sqrt2}{2} \frac{\sqrt3}{2} + \frac{\sqrt2}{2} \frac12 = \frac{ \sqrt6+\sqrt2 }{ 4 } \]
Answer: \[ \boxed{ \frac{ \sqrt6+\sqrt2 }{ 4 } } \]

Problem 9

Use Tangent Sum

Rewrite \(\tan(A+B)\) in terms of \(\tan A\) and \(\tan B\).

Step-by-Step Solution

Use the tangent sum identity. The numerator keeps the plus sign and the denominator switches to minus.

\[ \tan(A+B) = \frac{ \tan A+\tan B }{ 1-\tan A\tan B } \]
Answer: \[ \boxed{ \frac{ \tan A+\tan B }{ 1-\tan A\tan B } } \]

Level 4

Double, Half, and Power Reduction

Practice identities that come from angle multiplication and reduction.

Problem 10

Double Angle

Rewrite \[ 2\sin\theta\cos\theta \] as a single trig function.

Step-by-Step Solution

Recognize the sine double-angle identity.

\[ \sin(2\theta) = 2\sin\theta\cos\theta \]
Answer: \[ \boxed{\sin(2\theta)} \]

Problem 11

Double Angle

Rewrite \[ 2\cos^2\theta-1 \] as a single trig function.

Step-by-Step Solution

This is one of the three equivalent cosine double-angle forms.

\[ \cos(2\theta) = 2\cos^2\theta-1 \]
Answer: \[ \boxed{\cos(2\theta)} \]

Problem 12

Half Angle

Find the exact value of \[ \cos15^\circ \] using a half-angle identity.

Step-by-Step Solution

Since \(15^\circ = 30^\circ/2\), use the cosine half-angle identity. The answer is positive because \(15^\circ\) is in Quadrant I.

\[ \cos15^\circ = \sqrt{ \frac{ 1+\cos30^\circ }{ 2 } } \] \[ = \sqrt{ \frac{ 1+ \frac{\sqrt3}{2} }{ 2 } } = \frac{ \sqrt{ 2+\sqrt3 } }{ 2 } \]
Answer: \[ \boxed{ \frac{ \sqrt{ 2+\sqrt3 } }{ 2 } } \]

Problem 13

Power Reduction

Rewrite \[ \sin^2\theta \] using a power-reduction identity.

Step-by-Step Solution

Use the sine-squared power-reduction identity.

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

Level 5

Verify and Mix Identities

Choose your own path and transform one side until it matches the other.

Problem 14

Verify an Identity

Verify: \[ \tan\theta\cos\theta = \sin\theta. \]

Step-by-Step Solution

Work on the left side only and rewrite tangent.

\[ \tan\theta\cos\theta = \frac{ \sin\theta }{ \cos\theta } \cos\theta = \sin\theta \]
Verified: The left side simplifies to \(\sin\theta\).

Problem 15

Verify an Identity

Verify: \[ \frac{ 1-\cos^2\theta }{ \sin\theta } = \sin\theta. \]

Step-by-Step Solution

Use the Pythagorean identity on the numerator.

\[ \frac{ 1-\cos^2\theta }{ \sin\theta } = \frac{ \sin^2\theta }{ \sin\theta } = \sin\theta \]
Verified: The two sides match.

Problem 16

Mixed Identity

Simplify: \[ \frac{ 1-\cos(2\theta) }{ 2 } \cdot \csc^2\theta \]

Step-by-Step Solution

Use power reduction first.

\[ \frac{ 1-\cos(2\theta) }{ 2 } = \sin^2\theta \] \[ \sin^2\theta \cdot \csc^2\theta = \sin^2\theta \cdot \frac{ 1 }{ \sin^2\theta } = 1 \]
Answer: \[ \boxed{1} \]

Before You Finish

Trigonometric Identities Checklist

  1. Look for reciprocal, quotient, or Pythagorean forms first.
  2. Rewrite everything in sine and cosine when no clear identity stands out.
  3. For sum and difference identities, pay close attention to the sign pattern.
  4. Remember that double-angle formulas come from setting the two angles equal.
  5. Use the quadrant of \(\theta/2\) to choose the sign in half-angle identities.
  6. Use power reduction to replace squared trig functions with first-power expressions.
  7. When verifying an identity, work on one side at a time.

Continue Learning

Related Trigonometric Identities Resources

Review the full lesson, keep the identity reference sheet nearby, or continue strengthening the trig foundations these identities build on.

Trigonometric Identities Guide

Review all seven major identity families and worked examples.

Identity Reference Sheet

Keep the major identities together in one printable reference.

Unit Circle

Review exact sine and cosine values used in identity problems.

Special Right Triangles

Review exact values for 30-60-90 and 45-45-90 triangles.

SOH-CAH-TOA

Review the basic trig ratios behind sine, cosine, and tangent.

Trigonometry Resources

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