cos(x)tan^4(x)

Question

Use a power reducing identity to rewrite the following expression below in terms containing only first powers of cosine

 

Ive been working on this one for a minute and i keep getting lost

 

1/1+cos2x(1-2cos+1/2(1+cos4x)) this is as far as i can get and im honestly not sure that its completely correct. I feel like i missed some steps somewhere. Thank you for the help.

Answer 1

cos(x)tan⁴(x) = cos x * (sinx/cosx)⁴

                     = cos x *(sin⁴x/cos⁴x)

                     =sin⁴x / cos³x

                     =(sin²x)² / cos³x

                     = [(1-cos2x)/2]² / [(cos3x+3cosx)/4]

                     = [(1-cos2x)²/4 ] / [(cos3x+3cosx)/4]

                     =(1-cos2x)²  / (cos3x+3cosx)

                     =(1 + cos²2x - 2cos2x) /(cos3x+3cosx)

                     =[1+{ (cos4x+1)/2} - 2cos2x ] /  (cos3x+3cosx)

                     =( 2+ cos4x+ 1 - 4cos2x) / 2 (cos3x+3cosx)

                     =(cos4x - 4cos2x +3) / 2(cos3x+3cosx)