Trig identities worksheet for Precalculus

Precalculus identity questions use the sum and difference formulas for exact values at angles like 150° - 45°, the double angle formula to find tan 2θ from sin θ and cos θ, and the half angle formula for angles like 202.5°, half of 405°.

Also taught in Algebra 2, at its own level and with its own examples.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Trigonometric Identities Date Period Simplify or evaluate. Use the sum and difference identities $\sin ⁣ (120^{\circ} - 45^{\circ})$ $\cos ⁣ (150^{\circ} - 45^{\circ})$ $\sin ⁣ (60^{\circ} - 135^{\circ})$ $\cos ⁣ (120^{\circ} - 45^{\circ})$ $\cos ⁣ (60^{\circ} - 135^{\circ})$ $\sin ⁣ (45^{\circ} - 150^{\circ})$ $\sin ⁣ (225^{\circ} - 30^{\circ})$ $\sin ⁣ (135^{\circ} - 60^{\circ})$ $\cos ⁣ (225^{\circ} - 30^{\circ})$ Use the double-angle identities For an angle $θ$ in quadrant III with $\sin θ = - \frac{35}{37}$ , find $\sin 2 θ$ . Given that $\cos θ = - \frac{7}{25}$ , with $θ$ in quadrant II, find $\cos 2 θ$ . If $\tan θ = - \frac{21}{20}$ and $θ$ is in quadrant IV, find $\tan 2 θ$ . For an angle $θ$ in quadrant IV with $\tan θ = - \frac{4}{3}$ , find $\sin 2 θ$ . Given that $\sin θ = \frac{12}{13}$ , with $θ$ in quadrant II, find $\cos 2 θ$ . If $\cos θ = - \frac{11}{61}$ and $θ$ is in quadrant II, find $\tan 2 θ$ . Given that $\sin θ = - \frac{56}{65}$ , with $θ$ in quadrant III, find $\sin 2 θ$ . For an angle $θ$ in quadrant III with $\cos θ = - \frac{8}{17}$ , find $\tan 2 θ$ . If $\cos θ = - \frac{48}{73}$ and $θ$ is in quadrant II, find $\cos 2 θ$ . Find exact values with half angles Find $\tan {157.5}^{\circ}$ exactly Find $\cos {22.5}^{\circ}$ exactly Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Trigonometric Identities

Date Period

Simplify or evaluate.

Use the sum and difference identities
$\sin ⁣ (120^{\circ} - 45^{\circ})$
$\cos ⁣ (150^{\circ} - 45^{\circ})$
$\sin ⁣ (60^{\circ} - 135^{\circ})$
$\cos ⁣ (120^{\circ} - 45^{\circ})$
$\cos ⁣ (60^{\circ} - 135^{\circ})$
$\sin ⁣ (45^{\circ} - 150^{\circ})$
$\sin ⁣ (225^{\circ} - 30^{\circ})$
$\sin ⁣ (135^{\circ} - 60^{\circ})$
$\cos ⁣ (225^{\circ} - 30^{\circ})$

Use the double-angle identities
For an angle $θ$ in quadrant III with $\sin θ = - \frac{35}{37}$ , find $\sin 2 θ$ .
Given that $\cos θ = - \frac{7}{25}$ , with $θ$ in quadrant II, find $\cos 2 θ$ .
If $\tan θ = - \frac{21}{20}$ and $θ$ is in quadrant IV, find $\tan 2 θ$ .
For an angle $θ$ in quadrant IV with $\tan θ = - \frac{4}{3}$ , find $\sin 2 θ$ .
Given that $\sin θ = \frac{12}{13}$ , with $θ$ in quadrant II, find $\cos 2 θ$ .
If $\cos θ = - \frac{11}{61}$ and $θ$ is in quadrant II, find $\tan 2 θ$ .
Given that $\sin θ = - \frac{56}{65}$ , with $θ$ in quadrant III, find $\sin 2 θ$ .
For an angle $θ$ in quadrant III with $\cos θ = - \frac{8}{17}$ , find $\tan 2 θ$ .
If $\cos θ = - \frac{48}{73}$ and $θ$ is in quadrant II, find $\cos 2 θ$ .

Find exact values with half angles
Find $\tan {157.5}^{\circ}$ exactly
Find $\cos {22.5}^{\circ}$ exactly



Sigma Prep · sigmaprep.io/worksheets Name Trigonometric IdentitiesAnswer keyVersion 2 Date Period Simplify or evaluate. Use the sum and difference identities $\sin ⁣ (120^{\circ} - 45^{\circ})$ $\frac{\sqrt{6} + \sqrt{2}}{4}$ $\cos ⁣ (150^{\circ} - 45^{\circ})$ $- \frac{\sqrt{6} - \sqrt{2}}{4}$ $\sin ⁣ (60^{\circ} - 135^{\circ})$ $- \frac{\sqrt{6} + \sqrt{2}}{4}$ $\cos ⁣ (120^{\circ} - 45^{\circ})$ $\frac{\sqrt{6} - \sqrt{2}}{4}$ $\cos ⁣ (60^{\circ} - 135^{\circ})$ $\frac{\sqrt{6} - \sqrt{2}}{4}$ $\sin ⁣ (45^{\circ} - 150^{\circ})$ $- \frac{\sqrt{6} + \sqrt{2}}{4}$ $\sin ⁣ (225^{\circ} - 30^{\circ})$ $- \frac{\sqrt{6} - \sqrt{2}}{4}$ $\sin ⁣ (135^{\circ} - 60^{\circ})$ $\frac{\sqrt{6} + \sqrt{2}}{4}$ $\cos ⁣ (225^{\circ} - 30^{\circ})$ $- \frac{\sqrt{6} + \sqrt{2}}{4}$ Use the double-angle identities For an angle $θ$ in quadrant III with $\sin θ = - \frac{35}{37}$ , find $\sin 2 θ$ . $\frac{840}{1369}$ Given that $\cos θ = - \frac{7}{25}$ , with $θ$ in quadrant II, find $\cos 2 θ$ . $- \frac{527}{625}$ If $\tan θ = - \frac{21}{20}$ and $θ$ is in quadrant IV, find $\tan 2 θ$ . $\frac{840}{41}$ For an angle $θ$ in quadrant IV with $\tan θ = - \frac{4}{3}$ , find $\sin 2 θ$ . $- \frac{24}{25}$ Given that $\sin θ = \frac{12}{13}$ , with $θ$ in quadrant II, find $\cos 2 θ$ . $- \frac{119}{169}$ If $\cos θ = - \frac{11}{61}$ and $θ$ is in quadrant II, find $\tan 2 θ$ . $\frac{1320}{3479}$ Given that $\sin θ = - \frac{56}{65}$ , with $θ$ in quadrant III, find $\sin 2 θ$ . $\frac{3696}{4225}$ For an angle $θ$ in quadrant III with $\cos θ = - \frac{8}{17}$ , find $\tan 2 θ$ . $- \frac{240}{161}$ If $\cos θ = - \frac{48}{73}$ and $θ$ is in quadrant II, find $\cos 2 θ$ . $- \frac{721}{5329}$ Find exact values with half angles Find $\tan {157.5}^{\circ}$ exactly $1 - \sqrt{2}$ Find $\cos {22.5}^{\circ}$ exactly $\frac{\sqrt{2 + \sqrt{2}}}{2}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Trigonometric IdentitiesAnswer keyVersion 2

Date Period

Simplify or evaluate.

Use the sum and difference identities
$\sin ⁣ (120^{\circ} - 45^{\circ})$ $\frac{\sqrt{6} + \sqrt{2}}{4}$
$\cos ⁣ (150^{\circ} - 45^{\circ})$ $- \frac{\sqrt{6} - \sqrt{2}}{4}$
$\sin ⁣ (60^{\circ} - 135^{\circ})$ $- \frac{\sqrt{6} + \sqrt{2}}{4}$
$\cos ⁣ (120^{\circ} - 45^{\circ})$ $\frac{\sqrt{6} - \sqrt{2}}{4}$
$\cos ⁣ (60^{\circ} - 135^{\circ})$ $\frac{\sqrt{6} - \sqrt{2}}{4}$
$\sin ⁣ (45^{\circ} - 150^{\circ})$ $- \frac{\sqrt{6} + \sqrt{2}}{4}$
$\sin ⁣ (225^{\circ} - 30^{\circ})$ $- \frac{\sqrt{6} - \sqrt{2}}{4}$
$\sin ⁣ (135^{\circ} - 60^{\circ})$ $\frac{\sqrt{6} + \sqrt{2}}{4}$
$\cos ⁣ (225^{\circ} - 30^{\circ})$ $- \frac{\sqrt{6} + \sqrt{2}}{4}$

Use the double-angle identities
For an angle $θ$ in quadrant III with $\sin θ = - \frac{35}{37}$ , find $\sin 2 θ$ . $\frac{840}{1369}$
Given that $\cos θ = - \frac{7}{25}$ , with $θ$ in quadrant II, find $\cos 2 θ$ . $- \frac{527}{625}$
If $\tan θ = - \frac{21}{20}$ and $θ$ is in quadrant IV, find $\tan 2 θ$ . $\frac{840}{41}$
For an angle $θ$ in quadrant IV with $\tan θ = - \frac{4}{3}$ , find $\sin 2 θ$ . $- \frac{24}{25}$
Given that $\sin θ = \frac{12}{13}$ , with $θ$ in quadrant II, find $\cos 2 θ$ . $- \frac{119}{169}$
If $\cos θ = - \frac{11}{61}$ and $θ$ is in quadrant II, find $\tan 2 θ$ . $\frac{1320}{3479}$
Given that $\sin θ = - \frac{56}{65}$ , with $θ$ in quadrant III, find $\sin 2 θ$ . $\frac{3696}{4225}$
For an angle $θ$ in quadrant III with $\cos θ = - \frac{8}{17}$ , find $\tan 2 θ$ . $- \frac{240}{161}$
If $\cos θ = - \frac{48}{73}$ and $θ$ is in quadrant II, find $\cos 2 θ$ . $- \frac{721}{5329}$

Find exact values with half angles
Find $\tan {157.5}^{\circ}$ exactly $1 - \sqrt{2}$
Find $\cos {22.5}^{\circ}$ exactly $\frac{\sqrt{2 + \sqrt{2}}}{2}$

How to do these

If $\sin θ = \frac{3}{5}$ and $\cos θ = \frac{4}{5}$ , find $\tan 2 θ$

tan θ = sin θ ÷ cos θ = 3/4.
tan 2θ = 2 tan θ / (1 - tan²θ) = (3/2) ÷ (1 - 9/16).
(3/2) ÷ (7/16) = 24/7.

The answer is $\frac{24}{7}$ .

Where students go wrong

Doubling the tangent. tan 2θ is not 2 tan θ: with tan θ = 3/4 it is 24/7, not 3/2, because of the 1 - tan²θ in the denominator.

Also called angle sum identities or compound angle formulas.

What you can put on this worksheet

Sum and difference of two angles: $\cos ⁣ (60^{\circ} + 30^{\circ})$ → $0$
Double angle from a given ratio: Given that $\cos θ = \frac{12}{37}$ and $\sin θ = \frac{35}{37}$ , find $\cos 2 θ$ . → $- \frac{1081}{1369}$
From a given ratio: If $\cos θ = \frac{8}{17}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ → $\frac{5 \sqrt{34}}{34}$
From one ratio and the quadrant: $\cos θ = - \frac{15}{17}$ and $θ$ is in Quadrant II. Find $\sin 2 θ$ → $- \frac{240}{289}$
One angle split into a sum or difference: Without a calculator, find $\cos 165^{\circ}$ . Give an exact answer. → $- \frac{\sqrt{6} + \sqrt{2}}{4}$
From two ratios and their quadrants: $\cos a = \frac{8}{17}$ with $a$ in Quadrant I, and $\cos b = \frac{4}{5}$ with $b$ in Quadrant I. Find $\sin (a + b)$ → $\frac{84}{85}$
Exact values of half a special angle: Find $\sin 75^{\circ}$ exactly → $\frac{\sqrt{6} + \sqrt{2}}{4}$

Questions about these worksheets

Yes. Every trig identities sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the trig identities questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are two levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 7: sum and difference of two angles, double angle from a given ratio, from a given ratio, from one ratio and the quadrant, one angle split into a sum or difference, from two ratios and their quadrants and exact values of half a special angle. Tick as many as you want and set how many of each, or let it spread them evenly.

Assuming sine distributes. sin(A + B) is not sin A + sin B, and the gap between them is large: sin(45 + 45) is 1, while sin 45 + sin 45 is about 1.41. The formula exists precisely because the obvious thing is false.

Trig identities worksheet for Precalculus

Select your difficulty

Use the sum and difference identities

Use the double-angle identities

Find exact values with half angles

Use the sum and difference identities

Use the double-angle identities

Find exact values with half angles

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these trig identities worksheets free?+

Do the trig identities worksheets come with an answer key?+

Are these trig identities worksheets printable?+

What grade are these trig identities worksheets for?+

Can I get a different version of the same trig identities worksheet?+

Is this a free alternative to Kuta Software for trig identities?+

Can I choose how hard the trig identities questions are?+

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What do students get wrong with trig identities?+