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TrigonometryGrades 9–123 min read

Sum and Difference Identities

Sum and difference identities express trig values of combined angles using the values of the individual angles.

Cheat sheet
Trigonometric functions do not distribute over angle addition; their correct combination rules mix sine and cosine.

Sine identities

For any angles $A$ and $B$,

$$ \sin(A+B)=\sin A\cos B+\cos A\sin B, $$
$$ \sin(A-B)=\sin A\cos B-\cos A\sin B. $$

For sine, the sign in the formula matches the sign between the angles.

Cosine identities

The cosine rules are

$$ \cos(A+B)=\cos A\cos B-\sin A\sin B, $$
$$ \cos(A-B)=\cos A\cos B+\sin A\sin B. $$

Cosine uses the opposite sign: sum produces subtraction, while difference produces addition.

Tangent identities

When defined,

$$ \tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}, $$
$$ \tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}. $$

These follow by dividing sine's identity by cosine's identity. Denominators identify inputs where tangent is undefined.

Worked example: exact value of cosine 75°

Worked example: exact sine of 15°

Write $15^\circ=45^\circ-30^\circ$:

$$ \sin15^\circ =\sin45^\circ\cos30^\circ-\cos45^\circ\sin30^\circ =\frac{\sqrt6-\sqrt2}{4}. $$

Different angle decompositions can work, but choose special angles with known exact values.

Expanding expressions

The identities expand a compound angle into products:

$$ \sin(x+\pi/3) =\sin x\cos(\pi/3)+\cos x\sin(\pi/3) =\frac12\sin x+\frac{\sqrt3}{2}\cos x. $$

This form can support graphing, solving, or verification.

Compressing expressions

Read identities in reverse to combine expressions. For instance,

$$ \sin x\cos20^\circ+\cos x\sin20^\circ =\sin(x+20^\circ). $$

Coefficient patterns and signs reveal the matching identity.

Set $B=A$ in a sum identity to obtain double-angle identities. Set $A=0$ or replace an angle with its negative to derive even-odd relationships such as

$$ \sin(-x)=-\sin x,qquad \cos(-x)=\cos x. $$

The formulas form a connected system rather than an isolated list.

Applications to non-special angles

If trig values of $A$ and $B$ are known from triangles or quadrant information, the identities can find exact values for $A\pm B$. Determine missing sides, apply signs from quadrants, and then use the correct formula.

When only a trig value is given, the quadrant is necessary to determine the sign of the other ratios.

Formula memory and checks

For sine: “same sign.” For cosine: “opposite sign.” Tangent follows sine-like signs in the numerator and cosine-like opposite signs in the denominator.

Use benchmark estimates, quadrant signs, or a calculator decimal to check an exact result after the algebra is complete.

Common mistakes

Writing $\sin(A+B)=\sin A+\sin B$. Trig functions do not distribute over addition.

Using the wrong cosine sign. Cosine switches the operation sign.

Losing radicals or denominators. Substitute exact values with parentheses.

Choosing angles without known exact values. Decompose strategically.

Ignoring the compound angle's quadrant. Use it to check the final sign.

Quick self-check

  • Is the compound angle written as a useful sum or difference?
  • Did I choose the sine, cosine, or tangent formula accurately?
  • Are special-angle values exact?
  • Does the final sign match the compound angle's quadrant?
  • Can a decimal estimate confirm the magnitude without replacing the exact answer?
Check your understanding

Try it yourself

Hints are part of learning. Open one whenever it makes the next step feel possible.

1 practice question
Question 1Find an exact compound-angle value · Standard

What is the exact value of cos 75°?

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