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Unit 5 · Analytic Trigonometry
5.4 Sum and Difference
Identities
Objective: Use sum and difference formulas to evaluate trigonometric functions exactly and simplify trigonometric expressions into a single function.
\(\sin(\alpha \pm \beta)\) \(\cos(\alpha \pm \beta)\) \(\tan(\alpha \pm \beta)\) Exact values Simplify expressions
Motivation

Why these identities matter

Special angles are limited. You can find exact values like \(\sin 30^\circ = \dfrac{1}{2}\) and \(\sin 45^\circ = \dfrac{\sqrt{2}}{2}\) directly from the unit circle.

Not every angle is special. For example, \(\sin 195^\circ\) is not on the standard chart, but \(195^\circ = 150^\circ + 45^\circ\).

Sum and difference identities bridge that gap. They let you evaluate exact values and rewrite expressions such as \(\cos\left(\theta - \dfrac{3\pi}{2}\right)\) as a single trig function.

Reference

Sum and Difference Identities

Sin
\[\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta\]
Cos
\[\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta\]
Tan
\[\tan(\alpha \pm \beta) = \dfrac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}\]
For cosine and tangent, the sign in the denominator or second term flips relative to the sign inside the argument.
Example 1

Find the exact value

\[\cos 15^\circ\]
Rewrite the angle
\[\cos 15^\circ = \cos(45^\circ - 30^\circ)\]
Write \(15^\circ\) as a difference of special angles.
Apply the cosine difference identity
\[\cos(45^\circ - 30^\circ) = \cos45^\circ\cos30^\circ + \sin45^\circ\sin30^\circ\]
The sign flips for cosine: \(\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta\).
Answer
\[\cos 15^\circ = \dfrac{\sqrt{2}}{2}\cdot\dfrac{\sqrt{3}}{2} + \dfrac{\sqrt{2}}{2}\cdot\dfrac{1}{2} = \dfrac{\sqrt{6} + \sqrt{2}}{4}\]
Example 2

Find the exact value

\[\sin\left(\dfrac{\pi}{12}\right)\]
Rewrite the angle
\[\sin\left(\dfrac{\pi}{12}\right) = \sin\left(\dfrac{\pi}{3} - \dfrac{\pi}{4}\right)\]
\(\dfrac{\pi}{12}\) is the difference of two special angles.
Apply the sine difference identity
\[\sin\left(\dfrac{\pi}{3} - \dfrac{\pi}{4}\right) = \sin\dfrac{\pi}{3}\cos\dfrac{\pi}{4} - \cos\dfrac{\pi}{3}\sin\dfrac{\pi}{4}\]
For sine, the sign matches the sign inside the argument.
Answer
\[\sin\left(\dfrac{\pi}{12}\right) = \dfrac{\sqrt{3}}{2}\cdot\dfrac{\sqrt{2}}{2} - \dfrac{1}{2}\cdot\dfrac{\sqrt{2}}{2} = \dfrac{\sqrt{6} - \sqrt{2}}{4}\]
Example 3

Find the exact value

\[\tan 15^\circ\]
Rewrite the angle
\[\tan 15^\circ = \tan(45^\circ - 30^\circ)\]
Use a difference of special angles.
Apply the tangent difference identity
\[\tan(45^\circ - 30^\circ) = \dfrac{\tan45^\circ - \tan30^\circ}{1 + \tan45^\circ\tan30^\circ}\]
For \(\tan(\alpha - \beta)\), the denominator uses the opposite sign.
Substitute known values
\[\dfrac{1 - \dfrac{\sqrt{3}}{3}}{1 + \dfrac{\sqrt{3}}{3}}\]
\(\tan45^\circ = 1\) and \(\tan30^\circ = \dfrac{\sqrt{3}}{3}\).
Rationalize
\[\dfrac{1 - \dfrac{\sqrt{3}}{3}}{1 + \dfrac{\sqrt{3}}{3}} = \dfrac{3 - \sqrt{3}}{3 + \sqrt{3}} \cdot \dfrac{3 - \sqrt{3}}{3 - \sqrt{3}}\]
Multiply by the conjugate to clear the radical in the denominator.
Answer
\[\tan 15^\circ = \dfrac{(3 - \sqrt{3})^2}{9 - 3} = \dfrac{12 - 6\sqrt{3}}{6} = 2 - \sqrt{3}\]
Example 4

Find the exact value

\[\text{Given }\sin u = \dfrac{4}{5},\ 0 < u < \dfrac{\pi}{2},\ \cos v = -\dfrac{12}{13},\ \dfrac{\pi}{2} < v < \pi,\ \text{find }\sin(u+v)\]
Find \(\cos u\)
\[\cos u = \sqrt{1 - \sin^2u} = \sqrt{1 - \dfrac{16}{25}} = \dfrac{3}{5}\]
Since \(u\) is in Quadrant I, cosine is positive.
Find \(\sin v\)
\[\sin v = \sqrt{1 - \cos^2v} = \sqrt{1 - \dfrac{144}{169}} = \dfrac{5}{13}\]
Since \(v\) is in Quadrant II, sine is positive.
Apply the sum identity
\[\sin(u+v) = \sin u \cos v + \cos u \sin v\]
Now all four trig values are known.
Substitute
\[\sin(u+v) = \dfrac{4}{5}\left(-\dfrac{12}{13}\right) + \dfrac{3}{5}\cdot\dfrac{5}{13} = -\dfrac{48}{65} + \dfrac{15}{65}\]
Combine the two products over a common denominator.
Answer
\[\sin(u+v) = -\dfrac{33}{65}\]
Example 5

Simplify

\[\sin 220^\circ\cos 5^\circ + \cos 220^\circ\sin 5^\circ\]
Recognize the pattern
\[\sin\alpha\cos\beta + \cos\alpha\sin\beta = \sin(\alpha+\beta)\]
This expression already matches the sine sum identity.
Rewrite as one function
\[\sin 220^\circ\cos 5^\circ + \cos 220^\circ\sin 5^\circ = \sin(220^\circ + 5^\circ)\]
Collapse the pattern into a single sine expression.
Combine the angles
\[\sin(220^\circ + 5^\circ) = \sin 225^\circ\]
Now evaluate the special-angle result.
Answer
\[\sin 225^\circ = -\dfrac{\sqrt{2}}{2}\]
Example 6

Simplify

\[\cos\left(\theta - \dfrac{3\pi}{2}\right)\]
Apply the cosine difference identity
\[\cos\left(\theta - \dfrac{3\pi}{2}\right) = \cos\theta\cos\dfrac{3\pi}{2} + \sin\theta\sin\dfrac{3\pi}{2}\]
For cosine, subtracting inside becomes adding between the terms.
Use unit-circle values
\[\cos\theta\cdot 0 + \sin\theta\cdot(-1)\]
\(\cos\dfrac{3\pi}{2} = 0\) and \(\sin\dfrac{3\pi}{2} = -1\).
Answer
\[\cos\left(\theta - \dfrac{3\pi}{2}\right) = -\sin\theta\]
Example 7

Simplify

\[\tan(\theta + 3\pi)\]
Apply the tangent sum identity
\[\tan(\theta + 3\pi) = \dfrac{\tan\theta + \tan 3\pi}{1 - \tan\theta\tan 3\pi}\]
Start with the identity before using periodic values.
Substitute \(\tan 3\pi\)
\[\dfrac{\tan\theta + 0}{1 - \tan\theta\cdot 0}\]
\(\tan 3\pi = 0\) because \(3\pi\) lands on the x-axis.
Answer
\[\tan(\theta + 3\pi) = \tan\theta\]