Question

Those are not the powers question from transformation of product and sum trigonometry

Answer 1

Answer:

see explanation

Step-by-step explanation:

Using the sum to product identity

cosx + cosy = 2cos($\frac{x+y}{2}$ )cos( $\frac{x-y}{2}$ )

Consider left side

(cos5A +cos3A) + (cos15A + cos7A)

= 2cos($\frac{5A+3A}{2}$ )cos($\frac{5A-3A}{2}$ ) + 2cos( $\frac{15A+7A}{2}$) cos( $\frac{15A-7A}{2}$ )

= 2cos($\frac{8A}{2}$) cos($\frac{2A}{2}$) + 2cos($\frac{22A}{2}$ )cos($\frac{8A}{2}$ )

= 2cos4AcosA + 2cos11A cos4A ← factor out 2cos4A from both terms

= 2cos4A( cos11A + cosA) ← repeat the process

= 2cos4A( 2cos($\frac{11A+A}{2}$ )cos($\frac{11A-A}{2}$ )

= 2cos4A(2cos($\frac{12A}{2}$)cos($\frac{10A}{2}$ )

= 2cos4A(2cos6A cos5A)

= 4cos4Acos5Acos6A

= right side , thus verified