Question

find the exact value without a calculator. double - angle formulas: $\sin(2\theta)=2\sin\theta\cos\theta$ $\cos(2\theta)=\cos^{2}\theta - \sin^{2}\theta$ $\tan(2\theta)=\frac{2\tan\theta}{1 - \tan^{2}\theta}$ half - angle formulas: $\sin(\frac{\theta}{2})=\pm\sqrt{\frac{1 - \cos\theta}{2}}$ $\cos(\frac{\theta}{2})=\pm\sqrt{\frac{1 + \cos\theta}{2}}$ $\tan(\frac{\theta}{2})=\pm\sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}}$ $\cos\frac{\pi}{8}=\frac{\sqrt{?+\sqrt{\square}}}{\square}$

Explanation:

Step1: Use the half - angle formula for cosine

We know that \(\cos\frac{\theta}{2}=\pm\sqrt{\frac{1 + \cos\theta}{2}}\). Let \(\theta=\frac{\pi}{4}\), then \(\frac{\theta}{2}=\frac{\pi}{8}\).

Step2: Substitute \(\theta = \frac{\pi}{4}\) into the formula

Since \(\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\), we have \(\cos\frac{\pi}{8}=\sqrt{\frac{1+\cos\frac{\pi}{4}}{2}}\).

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Answer:

\(\cos\frac{\pi}{8}=\frac{\sqrt{2+\sqrt{2}}}{2}\), so the green box is \(2\), the white box is \(2\), and the gray box is \(2\)