Question

Use the power-reducing formulas to rewrite the expression in terms of first powers of the cosines of multiple angles.4 sin4(2x)

Answer 1

We have to reduce the expression:

`4\sin^4(2x)`

With power-reducing formulas we use identities that let us replace higher exponents terms with lower exponents terms.

We can start using the following identity:

`\sin^2\theta=(1)/(2)(1-\cos2\theta)`

Replacing in our formula we obtain:

`\begin{gathered} 4\sin^4(2x)=4[(1)/(2)(1-\cos(4x))]^2 \\ 4\sin^4(2x)=4\cdot((1)/(2))^2[1-\cos(4x)]^2 \\ 4\sin^4(2x)=4\cdot(1)/(4)\cdot(1^2-2\cos(4x)+\cos^2(4x)) \\ 4\sin^4(2x)=1-2\cos(4x)+\cos^2(4x) \end{gathered}`

We can replace the last term using this identity:

`\cos^2\theta=(1)/(2)(1+\cos2\theta)`

Then, we will obtain:

`\cos^2(4x)=(1)/(2)[1+\cos(8x)]`

Replacing in the equation we obtain:

`\begin{gathered} 1-2\cos(4x)+\cos^2(4x) \\ 1-2\cos(4x)+(1)/(2)(1+\cos(8x)) \\ 1-2\cos(4x)+(1)/(2)+(1)/(2)\cos(8x) \\ (3)/(2)-2\cos(4x)+(1)/(2)\cos(8x) \end{gathered}`

Answer:

3/2 - 2*cos(4x) + 1/2*cos(8x)