Worked example

Find the exact value of sin 15°

Answer

(√6 − √2)/4

15° = 45° − 30°
Difference of two special angles.
sin(45° − 30°) = sin 45° cos 30° − cos 45° sin 30°
Sine of a difference. · Sine of a difference
= (√2/2)(√3/2) − (√2/2)(1/2)
Special-angle values.
= (√6 − √2)/4
Combine.

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

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

Sine of a difference — Same family with a minus. It is the sum formula with B replaced by −B, so the second term changes sign — handy for exact values like sin 15° = sin(45° − 30°).
Sine of a sum — The angle itself is a sum and you need the functions separately — to evaluate exactly (sin 75° = sin(45° + 30°)), to expand, or to recognise the pattern running backwards.

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.