Question

How do I use the reducing formulas to rewrite the expression in terms of first powers of the cosines of multiple angle 5 cos^4(x) ?

Answer 1

`15/8+5/2cos2x+5/8cos4x`

Explanation:

Rewrite `5cos^4x` as `5(cos^2x)^2`.

Now, recall `cos^2x=1/2(1+cos2x)`

Apply this to the simplified expression:

`5(1/2(1+cos2x))^2=5/4(1+cos2x)^2=5/4(1+2cos2x+cos^2(2x))`

We will have to apply the power reduction formula again, to `cos^2(2x).` Let `u=2x:`

`cos^2(2x)=cos^2(u)=1/2(1+cos(2u))=1/2(1+cos(4x))`

So, we now have

`5/4(1+2cos2x+cos^2(2x))=5/4(1+2cos2x+1/2(1+cos4x))=5/4(1+2cos2x+1/2+1/2cos4x)=5/4+5/2cos2x+5/8+5/8cos4x=15/8+5/2cos2x+5/8cos4x`