`tan^2(2x)cos^4(2x)` Use the power reducing formulas to rewrite the expression in terms of the first power of the cosine.

It is known that

`cos^2(y)=(1/2)(1+cos(2y))`  and

`tan^2(y)=(1-cos(2y))/(1+cos(2y)).`

Repeating the first formula we obtain

`cos^4(y)=(1/4)(1+2cos(2y)+cos^2(2y))=`

`=(1/4)(1+2cos(2y)+(1/2)(1+cos(4y))=`

`=(1/8)(3+4cos(2y)+cos(4y)).`

Finally, for `y=2x` 

`tan^2(2x)*cos^4(2x)=(1-cos(4x))/(1+cos(4x))*(1/8)(3+4cos(4x)+cos(8x)).`

It is known that

`cos^2(y)=(1/2)(1+cos(2y))`  and

`tan^2(y)=(1-cos(2y))/(1+cos(2y)).`

Repeating the first formula we obtain

`cos^4(y)=(1/4)(1+2cos(2y)+cos^2(2y))=`

`=(1/4)(1+2cos(2y)+(1/2)(1+cos(4y))=`

`=(1/8)(3+4cos(2y)+cos(4y)).`

Finally, for `y=2x` 

`tan^2(2x)*cos^4(2x)=(1-cos(4x))/(1+cos(4x))*(1/8)(3+4cos(4x)+cos(8x)).`