Prove the following : 2+ 2+2+2 8 x =2 x - Vidyadip

Question

Prove the following : $\sqrt{2+\sqrt{2+\sqrt{2+2 \cos 8 x}}}=2$ $\cos x$

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Answer

$\begin{aligned}
& \text { L.H.S. }=\sqrt{2+\sqrt{2+\sqrt{2+2 \cos 8 x}}} \\
&=\sqrt{2+\sqrt{2+\sqrt{2[1+\cos 2(4 x)]}}} \\
&=\sqrt{2+\sqrt{2+\sqrt{2 \times 2 \cos ^2 4 x}}} \\
&=\sqrt{2+\sqrt{2+2 \cos 4 x}} \\
&=\sqrt{2+\sqrt{2[1+\cos 2(2 x)]}} \\
&=\sqrt{2+\sqrt{2 \times 2 \cos ^2 2 x}} \\
&=\sqrt{2+2 \cos ^{2 x}}=\sqrt{2(1+\cos 2 x)} \\
&=\sqrt{2 \times 2 \cos ^2 x} \\
&=2 \text { cos } x \\
&=\text { R.H.S. }
\end{aligned}$
[Note: The question has been modified.]

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