Write as an algebraic expression in x: cos (tan ^-1 x - sin ^-1 x)

sqrt(1 - x^2) + x^2sqrt(1 + x^2)

Verify: cos (x - 3π/2) = -sin x

cos (x - 3π/2); = cos xcos 3π/2 + sin xsin 3π/2 Sum and difference identity; = cos x × 0 + sin x × (-1) Exact values; = -sin x Algebra

Worksheets · Precalculus

Sum and difference identities worksheet for Precalculus

Precalculus works these in degrees and in radians: split an angle like 105° or 17π/12 into special angles, evaluate a sum or difference written out, or combine two ratios given with their quadrants. tan(a - b) needs both tangents found first.

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 Sum and Difference Identities Date Period Find each exact value, or answer as asked. Use the sum and difference identities $\sin ⁣ (120^{\circ} - 45^{\circ})$ $\cos a = \frac{5}{13}$ with $a$ in Quadrant I, and $\cos b = \frac{4}{5}$ with $b$ in Quadrant IV. Find $\sin (a + b)$ $\cos ⁣ (240^{\circ} + 30^{\circ})$ $\cos a = \frac{12}{13}$ with $a$ in Quadrant IV, and $\sin b = - \frac{8}{17}$ with $b$ in Quadrant III. Find $\tan (a - b)$ $\cos ⁣ (45^{\circ} - 150^{\circ})$ $\cos a = - \frac{5}{13}$ with $a$ in Quadrant III, and $\sin b = \frac{15}{17}$ with $b$ in Quadrant II. Find $\sin (a + b)$ $\sin ⁣ (300^{\circ} + 30^{\circ})$ $\sin a = - \frac{7}{25}$ with $a$ in Quadrant IV, and $\sin b = \frac{8}{17}$ with $b$ in Quadrant I. Find $\sin (a - b)$ $\cos ⁣ (225^{\circ} - 30^{\circ})$ $\sin a = - \frac{4}{5}$ with $a$ in Quadrant III, and $\cos b = \frac{24}{25}$ with $b$ in Quadrant IV. Find $\sin (a + b)$ $\sin ⁣ (270^{\circ} + 45^{\circ})$ $\cos a = - \frac{8}{17}$ with $a$ in Quadrant III, and $\cos b = - \frac{3}{5}$ with $b$ in Quadrant III. Find $\tan (a + b)$ $\cos ⁣ (135^{\circ} - 60^{\circ})$ Find exact values with angle sums Use a sum or difference identity to find the exact value of $\cot \frac{23 π}{12}$ . Split the angle into two special angles to find $\csc \frac{5 π}{12}$ exactly. Split the angle into two special angles to find $\tan ⁣ (- \frac{π}{12})$ exactly. Find the exact value of $\csc 165^{\circ}$ . Find the exact value of $\cot ⁣ (- \frac{11 π}{12})$ . Split the angle into two special angles to find $\sec 165^{\circ}$ exactly. Without a calculator, find $\sec \frac{17 π}{12}$ . Give an exact answer. Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Sum and Difference Identities

Date Period

Find each exact value, or answer as asked.

Use the sum and difference identities
$\sin ⁣ (120^{\circ} - 45^{\circ})$
$\cos a = \frac{5}{13}$ with $a$ in Quadrant I, and $\cos b = \frac{4}{5}$ with $b$ in Quadrant IV. Find $\sin (a + b)$
$\cos ⁣ (240^{\circ} + 30^{\circ})$
$\cos a = \frac{12}{13}$ with $a$ in Quadrant IV, and $\sin b = - \frac{8}{17}$ with $b$ in Quadrant III. Find $\tan (a - b)$
$\cos ⁣ (45^{\circ} - 150^{\circ})$
$\cos a = - \frac{5}{13}$ with $a$ in Quadrant III, and $\sin b = \frac{15}{17}$ with $b$ in Quadrant II. Find $\sin (a + b)$
$\sin ⁣ (300^{\circ} + 30^{\circ})$
$\sin a = - \frac{7}{25}$ with $a$ in Quadrant IV, and $\sin b = \frac{8}{17}$ with $b$ in Quadrant I. Find $\sin (a - b)$
$\cos ⁣ (225^{\circ} - 30^{\circ})$
$\sin a = - \frac{4}{5}$ with $a$ in Quadrant III, and $\cos b = \frac{24}{25}$ with $b$ in Quadrant IV. Find $\sin (a + b)$
$\sin ⁣ (270^{\circ} + 45^{\circ})$
$\cos a = - \frac{8}{17}$ with $a$ in Quadrant III, and $\cos b = - \frac{3}{5}$ with $b$ in Quadrant III. Find $\tan (a + b)$
$\cos ⁣ (135^{\circ} - 60^{\circ})$

Find exact values with angle sums
Use a sum or difference identity to find the exact value of $\cot \frac{23 π}{12}$ .
Split the angle into two special angles to find $\csc \frac{5 π}{12}$ exactly.
Split the angle into two special angles to find $\tan ⁣ (- \frac{π}{12})$ exactly.
Find the exact value of $\csc 165^{\circ}$ .
Find the exact value of $\cot ⁣ (- \frac{11 π}{12})$ .
Split the angle into two special angles to find $\sec 165^{\circ}$ exactly.
Without a calculator, find $\sec \frac{17 π}{12}$ . Give an exact answer.



Sigma Prep · sigmaprep.io/worksheets Name Sum and Difference IdentitiesAnswer keyVersion 2 Date Period Find each exact value, or answer as asked. Use the sum and difference identities $\sin ⁣ (120^{\circ} - 45^{\circ})$ $\frac{\sqrt{6} + \sqrt{2}}{4}$ $\cos a = \frac{5}{13}$ with $a$ in Quadrant I, and $\cos b = \frac{4}{5}$ with $b$ in Quadrant IV. Find $\sin (a + b)$ $\frac{33}{65}$ $\cos ⁣ (240^{\circ} + 30^{\circ})$ $0$ $\cos a = \frac{12}{13}$ with $a$ in Quadrant IV, and $\sin b = - \frac{8}{17}$ with $b$ in Quadrant III. Find $\tan (a - b)$ $- \frac{171}{140}$ $\cos ⁣ (45^{\circ} - 150^{\circ})$ $- \frac{\sqrt{6} - \sqrt{2}}{4}$ $\cos a = - \frac{5}{13}$ with $a$ in Quadrant III, and $\sin b = \frac{15}{17}$ with $b$ in Quadrant II. Find $\sin (a + b)$ $\frac{21}{221}$ $\sin ⁣ (300^{\circ} + 30^{\circ})$ $- \frac{1}{2}$ $\sin a = - \frac{7}{25}$ with $a$ in Quadrant IV, and $\sin b = \frac{8}{17}$ with $b$ in Quadrant I. Find $\sin (a - b)$ $- \frac{297}{425}$ $\cos ⁣ (225^{\circ} - 30^{\circ})$ $- \frac{\sqrt{6} + \sqrt{2}}{4}$ $\sin a = - \frac{4}{5}$ with $a$ in Quadrant III, and $\cos b = \frac{24}{25}$ with $b$ in Quadrant IV. Find $\sin (a + b)$ $- \frac{3}{5}$ $\sin ⁣ (270^{\circ} + 45^{\circ})$ $- \frac{\sqrt{2}}{2}$ $\cos a = - \frac{8}{17}$ with $a$ in Quadrant III, and $\cos b = - \frac{3}{5}$ with $b$ in Quadrant III. Find $\tan (a + b)$ $- \frac{77}{36}$ $\cos ⁣ (135^{\circ} - 60^{\circ})$ $\frac{\sqrt{6} - \sqrt{2}}{4}$ Find exact values with angle sums Use a sum or difference identity to find the exact value of $\cot \frac{23 π}{12}$ . $- 2 - \sqrt{3}$ Split the angle into two special angles to find $\csc \frac{5 π}{12}$ exactly. $\sqrt{6} - \sqrt{2}$ Split the angle into two special angles to find $\tan ⁣ (- \frac{π}{12})$ exactly. $\sqrt{3} - 2$ Find the exact value of $\csc 165^{\circ}$ . $\sqrt{6} + \sqrt{2}$ Find the exact value of $\cot ⁣ (- \frac{11 π}{12})$ . $2 + \sqrt{3}$ Split the angle into two special angles to find $\sec 165^{\circ}$ exactly. $\sqrt{2} - \sqrt{6}$ Without a calculator, find $\sec \frac{17 π}{12}$ . Give an exact answer. $- \sqrt{6} - \sqrt{2}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Sum and Difference IdentitiesAnswer keyVersion 2

Date Period

Find each exact value, or answer as asked.

Use the sum and difference identities
$\sin ⁣ (120^{\circ} - 45^{\circ})$ $\frac{\sqrt{6} + \sqrt{2}}{4}$
$\cos a = \frac{5}{13}$ with $a$ in Quadrant I, and $\cos b = \frac{4}{5}$ with $b$ in Quadrant IV. Find $\sin (a + b)$ $\frac{33}{65}$
$\cos ⁣ (240^{\circ} + 30^{\circ})$ $0$
$\cos a = \frac{12}{13}$ with $a$ in Quadrant IV, and $\sin b = - \frac{8}{17}$ with $b$ in Quadrant III. Find $\tan (a - b)$ $- \frac{171}{140}$
$\cos ⁣ (45^{\circ} - 150^{\circ})$ $- \frac{\sqrt{6} - \sqrt{2}}{4}$
$\cos a = - \frac{5}{13}$ with $a$ in Quadrant III, and $\sin b = \frac{15}{17}$ with $b$ in Quadrant II. Find $\sin (a + b)$ $\frac{21}{221}$
$\sin ⁣ (300^{\circ} + 30^{\circ})$ $- \frac{1}{2}$
$\sin a = - \frac{7}{25}$ with $a$ in Quadrant IV, and $\sin b = \frac{8}{17}$ with $b$ in Quadrant I. Find $\sin (a - b)$ $- \frac{297}{425}$
$\cos ⁣ (225^{\circ} - 30^{\circ})$ $- \frac{\sqrt{6} + \sqrt{2}}{4}$
$\sin a = - \frac{4}{5}$ with $a$ in Quadrant III, and $\cos b = \frac{24}{25}$ with $b$ in Quadrant IV. Find $\sin (a + b)$ $- \frac{3}{5}$
$\sin ⁣ (270^{\circ} + 45^{\circ})$ $- \frac{\sqrt{2}}{2}$
$\cos a = - \frac{8}{17}$ with $a$ in Quadrant III, and $\cos b = - \frac{3}{5}$ with $b$ in Quadrant III. Find $\tan (a + b)$ $- \frac{77}{36}$
$\cos ⁣ (135^{\circ} - 60^{\circ})$ $\frac{\sqrt{6} - \sqrt{2}}{4}$

Find exact values with angle sums
Use a sum or difference identity to find the exact value of $\cot \frac{23 π}{12}$ . $- 2 - \sqrt{3}$
Split the angle into two special angles to find $\csc \frac{5 π}{12}$ exactly. $\sqrt{6} - \sqrt{2}$
Split the angle into two special angles to find $\tan ⁣ (- \frac{π}{12})$ exactly. $\sqrt{3} - 2$
Find the exact value of $\csc 165^{\circ}$ . $\sqrt{6} + \sqrt{2}$
Find the exact value of $\cot ⁣ (- \frac{11 π}{12})$ . $2 + \sqrt{3}$
Split the angle into two special angles to find $\sec 165^{\circ}$ exactly. $\sqrt{2} - \sqrt{6}$
Without a calculator, find $\sec \frac{17 π}{12}$ . Give an exact answer. $- \sqrt{6} - \sqrt{2}$

How to do these

Find the exact value:
$\cos \frac{7 π}{12}$

7π/12 is π/3 + π/4.
cos(A + B) = cos A cos B - sin A sin B = (1/2)(√2/2) - (√3/2)(√2/2).

The answer is $\frac{\sqrt{2} - \sqrt{6}}{4}$ .

Where students go wrong

Mixing up the signs. cos(A + B) has a minus between the products and cos(A - B) a plus, the opposite of sine, so copying the sine pattern gives the right numbers with the wrong sign.

Also called angle sum formulas or compound angle identities.

What you can put on this worksheet

Sum and difference of two angles: $\cos ⁣ (60^{\circ} + 30^{\circ})$ → $0$
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 $\sin (a + b)$ → $\frac{84}{85}$
Negative angles: Use a sum or difference identity to find the exact value:
$\cos ⁣ (- \frac{17 π}{12})$ → $\frac{\sqrt{2} - \sqrt{6}}{4}$
Recognize the formula, then evaluate: Find the exact value:
$\cos 250^{\circ} \cos 10^{\circ} + \sin 250^{\circ} \sin 10^{\circ}$ → $- \frac{1}{2}$
Write as one trig function: Write as one trig function:
$\cos 4 θ \cos θ + \sin 4 θ \sin θ$ → $\cos 3 θ$
Inverse functions inside, as an algebraic expression: Write as an algebraic expression in $x$ : $\cos ⁣ (\tan^{- 1} x - \sin^{- 1} x)$ → $\frac{\sqrt{1 - x^{2}} + x^{2}}{\sqrt{1 + x^{2}}}$
Verify a shift identity: Verify:
$\cos ⁣ (x - \frac{3 π}{2}) = - ⁣ \sin x$
→ Answer
$\begin{aligned} \cos ⁣ (x - \frac{3 π}{2}) \\ = \cos x \cos \frac{3 π}{2} + \sin x \sin \frac{3 π}{2} & Sum and difference identity \\ = \cos x \cdot 0 + \sin x \cdot (- 1) & Exact values \\ = - ⁣ \sin x & Algebra \end{aligned}$

Questions about these worksheets

Yes. Every sum and difference 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 sum and difference 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 three 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 8: sum and difference of two angles, one angle split into a sum or difference, from two ratios and their quadrants, negative angles, recognize the formula, then evaluate, write as one trig function, inverse functions inside, as an algebraic expression and verify a shift identity. Tick as many as you want and set how many of each, or let it spread them evenly.

Assuming sine distributes over addition. sin(A + B) is not sin A + sin B, and the formula exists precisely because the obvious thing is false.

Sum and difference identities worksheet for Precalculus

Select your difficulty

Use the sum and difference identities

Find exact values with angle sums

Use the sum and difference identities

Find exact values with angle sums

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these sum and difference identities worksheets free?+

Do the sum and difference identities worksheets come with an answer key?+

Are these sum and difference identities worksheets printable?+

What grade are these sum and difference identities worksheets for?+

Can I get a different version of the same sum and difference identities worksheet?+

Is this a free alternative to Kuta Software for sum and difference identities?+

Can I choose how hard the sum and difference identities questions are?+

What kinds of question are on a sum and difference identities worksheet?+

What do students get wrong with sum and difference identities?+