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

Use The Powerreducing Formulas To Rewrite The Expression In Terms Of First Powers Of The Cosines Of Multiple Anglestan44x class=

Sagot :

Solution

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

The answer is:

[tex]\frac{3-4\cos(8x)+\cos(16x)}{3+4\cos(8x)+\cos(16x)}[/tex]

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