Answer to Question #301958 in Trigonometry for Weng

Question

Show the derivation of the formula.

Cos (u+v)=cos u cos v - sin u sin v

Answer 1

Derivation of the formula.

Cos (u+v)=cos u cos v - sin u sin v

(QP)² = (cos U - Cos V)² + (sin U - sin V)²

= cos² u - 2 cos u cos v + cos²v + sin²u -2 sin u sin v +sin²v

= 2 - 2(cos u cos v +sin u sin v) ...................................................(eq1)

(QP)² = 1² + 1² -2 *1 *1 * cos (u-v)....(cosine formula)

= 2 - 2 cos (u -v)............................................................................(eq2)

Equating eq1 and eq2 we get,

cos (u - v) = cos u cos v +sin u sin v

Let v = -v

cos(u−(−v))=cos u cos(−v) + sinu sin(−v)

(since cos (-a) = cos a and sin (-a) = - sin a)

hence

cos (u + v) = cos u cos v - sin u sin v [Answer]