Question

How do you prove cosθ + cos2θ + cos3θ = (2cosθ + 1) cos2θ ?

Answer 1

See explanation

Explanation:

We want to verify the identity

`cos(theta)+cos(2theta)+cos(3theta)=(2cos(theta)+1)cos(2theta)`

We will use the identity

• `cos(a)cos(b)=1/2(cos(a-b)+cos(a+b))`

Thus

`RHS=(2cos(theta)+1)cos(2theta)`

`color(white)(LHS)=2cos(2theta)cos(theta)+cos(2theta)`

`color(white)(LHS)=2(1/2((cos(2theta-theta)+cos(2theta+theta))))+cos(2theta)`

`color(white)(LHS)=cos(2theta-theta)+cos(2theta+theta)+cos(2theta)`

`color(white)(LHS)=cos(theta)+cos(3theta)+cos(2theta)=LHS`