Worked example

Find the exact value of cos 15° without a half-angle formula

Answer

(√6 + √2)/4

15° = 45° − 30°
Same split as the sine example.
cos(45° − 30°) = cos 45° cos 30° + sin 45° sin 30°
Cosine of a difference — note the plus. · Cosine of a difference
= (√2/2)(√3/2) + (√2/2)(1/2)
Values.
= (√6 + √2)/4
Combine — and note it equals sin 75°, as the cofunction pair predicts. · Cofunction identity for sine

cos(A − B) = cos A cos B + sin A sin B

Cosine of a difference — The minus-inside case for cosine — and the origin of the law of cosines when you expand c² = |A − B|² with this formula.

Open the verifier with the main identity pre-filled — substituting your own numbers into a step is the fastest way to find out which line you did not follow.