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I did $$\cos\left(\frac{45^\circ}{2}\right) = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$$ and ended by getting $\sqrt{\frac{2 + \sqrt{2}}{4}}$. But the answer in the book is $\frac{\sqrt{2 + \sqrt{2}}}{2}$.

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1 Answer

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$$\frac1{\sqrt2}=\cos\frac\pi4=\cos\left(2\frac\pi8\right)=\cos^2\frac\pi8-\sin^2\frac\pi8=2\cos^2\frac\pi8-1\implies$$

$$\cos\frac\pi8=\frac1{\sqrt2}\sqrt{\frac1{\sqrt2}+1}$$

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