Question

How do you prove `1-cos5thetacos3theta-sin5thetasin3theta=2sin^2theta`?

Answer 1

We use the identity `cos2theta=1-2sin^2theta` and `cos(A-B)=cosAcosB+sinAsinB`

Explanation:

We know that `cos2theta=1-2sin^2theta`

But as `2theta=5theta-3theta`, we have

`cos(5theta-3theta)=1-2sin^2theta`

or `cos5thetacos3theta+sin5thetasin3theta=1-2sin^2theta`

or `1-cos5thetacos3theta-sin5thetasin3theta=2sin^2theta`