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Chapter 6: Problem 36

Find the exact value of the expression. $$\cos \frac{\pi}{16} \cos \frac{3 \pi}{16}-\sin \frac{\pi}{16} \sin \frac{3 \pi}{16}$$

Short Answer

Expert verified
The exact value of the expression is \(\frac{\sqrt{2}}{2}\).

Step by step solution

01

Identify the Identity

In this step, identify the cosine of a sum of two angles identity: \(\cos(A+B) = \cos A \cos B - \sin A \sin B\). The given expression fits this formula with \(A = \frac{\pi}{16}\) and \(B = \frac{3\pi}{16}\)
02

Apply the Identity

Apply the formula \(\cos(A+B) = \cos A \cos B - \sin A \sin B\). So, \(\cos \frac{\pi}{16} \cos \frac{3 \pi}{16}-\sin \frac{\pi}{16} \sin \frac{3\pi}{16}\) simplifies to \(\cos(\frac{\pi}{16} + \frac{3\pi}{16})\)
03

Simplify the Expression

Simplify \(\cos(\frac{\pi}{16} + \frac{3\pi}{16})\) to \(\cos \frac{4\pi}{16}\), which further simplifies to \(\cos \frac{\pi}{4}\)
04

Calculate the Exact Value

Now calculate the exact value of \(\cos \frac{\pi}{4}\), which is \(\frac{\sqrt{2}}{2}\). Hence, the exact value of the given expression is \(\frac{\sqrt{2}}{2}\)

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