Question

Use the power-reducing formulas to rewrite the expression in terms of first powers of the cosines of multiple angles.tan4(4x)

Answer 1

`\begin{gathered} \tan^2(4x)=(1-\cos(8x))/(1+\cos(8x)) \\ \\ \Rightarrow\tan^4(4x)=(1-2\cos(8x)+\cos^2(8x))/(1+2\cos(8x)+\cos^2(8x)) \\ \\ \text{ since }\cos^2(8x)=(1+\cos(16x))/(2) \\ \\ \Rightarrow\tan^4(4x)=(1-2\cos(8x)+(1+\cos(16x))/(2))/(1+2\cos(8x)+(1+\cos(16x))/(2)) \\ \\ \Rightarrow\tan^4(4x)=(2-4\cos(8x)+1+\cos(16x))/(2+4\cos(8x)+1+\cos(16x)) \\ \\ \Rightarrow\tan^4(4x)=(3-4\cos(8x)+\cos(16x))/(3+4\cos(8x)+\cos(16x)) \end{gathered}`

The answer is:

`(3-4\cos(8x)+\cos(16x))/(3+4\cos(8x)+\cos(16x))`