Sum and difference formulas

These formulas find the ratios of a sum or difference of two angles:

• sin(A ± B) = sin A cos B ± cos A sin B: sin and cos alternate, and the sign stays the same.
• cos(A ± B) = cos A cos B ∓ sin A sin B: cos with cos, sin with sin, and the sign flips.
• tan(A + B) = (tan A + tan B)/(1 – tan A tan B) and tan(A – B) = (tan A – tan B)/(1 + tan A tan B).

The cos formula is proved on the unit circle by showing two chords are equal; all the others follow from it. Two useful special cases are cos(π/2 – x) = sin x and sin(π/2 – x) = cos x.

Exact values for new angles: split them into known angles. cos 105° = cos(60° + 45°) = cos 60° cos 45° – sin 60° sin 45° = (1/2)(1/√2) – (√3/2)(1/√2) = (1 – √3)/(2√2). Likewise tan 75° = tan(45° + 30°) = (1 + 1/√3)/(1 – 1/√3) = 2 + √3.

Allied angles come out of the same formulas: cos(π – x) = –cos x, sin(π – x) = sin x, cos(π + x) = –cos x, sin(π + x) = –sin x, cos(2π – x) = cos x and sin(2π – x) = –sin x.

Using given ratios. If sin A = 4/5 and cos B = 5/13, both in the first quadrant, first find cos A = 3/5 and sin B = 12/13. Then:

• sin(A + B) = (4/5)(5/13) + (3/5)(12/13) = 20/65 + 36/65 = 56/65.
• cos(A + B) = (3/5)(5/13) – (4/5)(12/13) = 15/65 – 48/65 = –33/65, so A + B is beyond 90°.

Proving with a split angle. Show that tan 50° = tan 40° + 2 tan 10°.

• Write 50° = 40° + 10°: tan 50° = (tan 40° + tan 10°)/(1 – tan 40° tan 10°).
• Cross-multiply: tan 50° – tan 50° tan 40° tan 10° = tan 40° + tan 10°.
• Since 50° + 40° = 90°, tan 50° tan 40° = 1. So tan 50° – tan 10° = tan 40° + tan 10°, which gives the result.

The same idea proves a classic identity: from tan 3x = tan(2x + x), cross-multiplying gives tan 3x tan 2x tan x = tan 3x – tan 2x – tan x.