Rewrite the given function as an equivalent function contain

Rewrite the given function as an equivalent function containing only cosine terms raised to a power of 1. f(x) = - 6 sin^2 (x) cos^2 (x) f(x) = (Type an exact answer in simplified form. Use integers or fractions for any numbers in the expressions. Do not factor.)

Solution

given f(X)=-6sin^2(x)cos^2(x)

Rearrange and use double angle formula for sine:

sin^2(x)cos^2(x) = ( (1/2)*( 2*sin(x)cos(x) ) )^2

= ( (1/2)*( sin(2x) ) )^2

= (1/4) * sin^2(2x)

Use power reducing formula for sine:

sin^2(x)cos^2(x)= (1/4) *(1/2)*( 1-cos(4x))

sin^2(x)cos^2(x)= ( 1 - cos(4x) ) / 8

f(x)=-6sin^2(x)cos^2(x)=-6( 1 - cos(4x) ) / 8

=-3(( 1 - cos(4x) ) / 4