Sum and Difference Identities

The sum and difference identities let you find the sine, cosine, or tangent of an angle written as the sum or difference of two other angles — without needing a calculator. They break a single unfamiliar angle like 75° into two familiar ones, such as 45° + 30°.

For sine, the sum and difference identities share the same pattern: sin(a ± b) = sin a cos b ± cos a sin b. Cosine flips the sign: cos(a + b) subtracts (cos a cos b − sin a sin b), while cos(a − b) adds (cos a cos b + sin a sin b). Getting the signs right is the most common sticking point, so it helps to practice each one from a table like the one above.

The tangent sum and difference identity comes from dividing the sine identity by the cosine identity: tan(a ± b) = (tan a ± tan b) / (1 ∓ tan a tan b). It uses the quotient identity for tangent as its starting point.

Find sin 75° using sin(45° + 30°). Substitute known values: sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2) / 4. This exact value would be hard to find any other way.

Sum and difference identities are the foundation for the double-angle identities, which are just the special case where a and b are equal.

Related lessons

• Trigonometry