Practice Pythagorean Trigonometric Identities in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The fundamental identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 and its derived forms: 1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta and 1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta.

On the unit circle, the point (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta) is always at distance 1 from the origin. By the Pythagorean theorem, x2+y2=1x^2 + y^2 = 1 becomes cos⁡2θ+sin⁡2θ=1\cos^2\theta + \sin^2\theta = 1. This single fact—that sine and cosine are tied to a circle—generates all three Pythagorean identities. Dividing through by cos⁡2θ\cos^2\theta or sin⁡2θ\sin^2\theta produces the other two forms.

Showing a random 20 of 50 problems.

Example 1

hard
Simplify sin⁡6x+cos⁡6x\sin^6 x + \cos^6 x in terms of sin⁡2xcos⁡2x\sin^2 x\cos^2 x.

Example 2

easy
Fill in: sin⁡217°+cos⁡217°=\sin^2 17° + \cos^2 17° = ___.

Example 3

easy
Simplify 1−cos⁡2θ1 - \cos^2\theta.

Example 4

hard
Solve 2cos⁡2x−1=sin⁡x2\cos^2 x - 1 = \sin x on [0,2π)[0, 2\pi).

Example 5

hard
Verify the identity 1+sin⁡θcos⁡θ+cos⁡θ1+sin⁡θ=2sec⁡θ\dfrac{1 + \sin\theta}{\cos\theta} + \dfrac{\cos\theta}{1 + \sin\theta} = 2\sec\theta.

Example 6

easy
If sin⁡(θ)=35\sin(\theta) = \frac{3}{5} and θ\theta is in Quadrant I, find cos⁡(θ)\cos(\theta) using the Pythagorean identity.

Example 7

hard
Verify (sin⁡x+cos⁡x)2=1+2sin⁡xcos⁡x(\sin x + \cos x)^2 = 1 + 2\sin x\cos x.

Example 8

medium
If tan⁡θ=−34\tan\theta = -\tfrac{3}{4} and θ\theta is in Quadrant II, find sec⁡θ\sec\theta and sin⁡θ\sin\theta.

Example 9

medium
Simplify sin⁡xcot⁡x\sin x\cot x.

Example 10

medium
Simplify tan⁡θ⋅cos⁡θ\tan\theta \cdot \cos\theta.

Example 11

easy
If cos⁡θ=12\cos\theta = \tfrac{1}{2}, find sin⁡2θ\sin^2\theta.

Example 12

medium
Simplify sin⁡2xsec⁡2x+sin⁡2x\sin^2 x\sec^2 x + \sin^2 x.

Example 13

medium
Simplify cos⁡2x−1cos⁡x−1\dfrac{\cos^2 x - 1}{\cos x - 1}.

Example 14

challenge
Express sin⁡4θ+cos⁡4θ\sin^4\theta + \cos^4\theta in terms of sin⁡2θcos⁡2θ\sin^2\theta\cos^2\theta, then simplify.

Example 15

easy
Fill in: sec⁡2θ=1+\sec^2\theta = 1 + ___.

Example 16

easy
State the identity relating tan⁡θ\tan\theta and sec⁡θ\sec\theta.

Example 17

easy
Simplify csc⁡2θ−cot⁡2θ\csc^2\theta - \cot^2\theta.

Example 18

easy
If sin⁡θ=0\sin\theta = 0, what is cos⁡2θ\cos^2\theta?

Example 19

challenge
Prove the identity 11−sin⁡θ+11+sin⁡θ=2sec⁡2θ\frac{1}{1 - \sin\theta} + \frac{1}{1 + \sin\theta} = 2\sec^2\theta.

Example 20

easy
Simplify csc⁡2x−1\csc^2 x - 1.
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