Triple-angle identities
sin 3A = 3sin A − 4sin³A cos 3A = 4cos³A − 3cos A tan 3A = (3tan A − tan³A) / (1 − 3tan²A)
Double-angle identities
sin 2A = 2 sin A cos A cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A tan 2A = 2tan A / (1 − tan²A)
Half-angle identities
sin(A/2) = √((1 − cos A) / 2) cos(A/2) = √((1 + cos A) / 2) tan(A/2) = √((1 − cos A) / (1 + cos A)) = sin A / (1 + cos A) = (1 − cos A) / sin A
Product identities
sin θ · sin 2θ · sin 4θ = (1/4) sin 4θ sin θ · cos θ · cos 2θ = (1/4) sin 4θ cos θ · cos 2θ · cos 4θ = sin 8θ / (8 sin θ) tan 20° · tan 40° · tan 80° = tan 60° = √3
Product-to-sum identities
sin A · sin B = (1/2) [cos(A − B) − cos(A + B)] cos A · cos B = (1/2) [cos(A − B) + cos(A + B)] sin A · cos B = (1/2) [sin(A + B) + sin(A − B)] cos A · sin B = (1/2) [sin(A + B) − sin(A − B)]
Sum-to-product identities
sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2) sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2) cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2) cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2)
Sum formulas in a triangle
When A + B + C = 180°:
tan A + tan B + tan C = tan A · tan B · tan C cot A·cot B + cot B·cot C + cot C·cot A = 1
Difference of cosines
cos C − cos D = −2 sin((C + D)/2) · sin((C − D)/2)