Find the values of : 8 - Vidyadip

Question

Find the values of : $\frac{\pi}{8}$

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Answer

We know that, $\cos ^2 \theta=\frac{1+\cos 2 \theta}{2}$
Substituting $\theta=\frac{\pi}{8}$, we get
$\begin{aligned}
\cos ^2 \frac{\pi}{8} & =\frac{1+\cos \frac{\pi}{4}}{2} \\
& =\frac{1+\frac{1}{\sqrt{2}}}{2}=\frac{\sqrt{2}+1}{2 \sqrt{2}} \\
\therefore \quad \cos \frac{\pi}{8} & =\sqrt{\frac{\sqrt{2}+1}{2 \sqrt{2}}} \quad \cdots\left[\because \cos \frac{\pi}{8} \text { is positive }\right] \\
\therefore \quad \cos \frac{\pi}{8} & =\sqrt{\frac{\sqrt{2}+1}{2 \sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}} \\
& =\sqrt{\frac{2+\sqrt{2}}{4}}\\
\therefore \quad \cos \frac{\pi}{8} & =\frac{\sqrt{2+\sqrt{2}}}{2}
\end{aligned}$

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