Question

Prove the equation cos(a+b).cos(a-b)=cos^2a - sin^2b. Show your solution.​

Answer 1

Answer:

See proof below

Explanation:

We need

(x+y)(x−y)=x2−y2

cos(a+b)=cosacosb−sinasinb

cos(a−b)=cosacosb+sinasinb

cos2a+sin2a=1

cos2b+sin2b=1

Therefore,

LHS=cos(a+b)cos(a−b)

=(cosacosb−sinasinb)(cosacosb+sinasinb)

=cos2acos2b−sin2asin2b

=cos2b(1−sin2a)−sin2a(1−cos2b)

=cos2b−sin2a

= RHS