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Chapter 5: 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
\(\cos(\frac{\pi}{4})\) is the exact value of the expression, which simplifies to \(\frac{\sqrt{2}}{2}\).

Step by step solution

01

Recall Cosine Subtraction Identity

The cosine subtraction formula states that \(\cos(A - B) = \cos A \cos B + \sin A \sin B\). So, if there's a minus instead of a plus in the equation, it can be interpreted as \(\cos(A + B).\)
02

Substitute Given Values

Given \(A = \frac{\pi}{16} \) and \(B = \frac{3\pi}{16}\), the expression then becomes \(\cos(A + B) = \cos \frac{\pi}{16} \cos \frac{3 \pi}{16} - \sin \frac{\pi}{16} \sin \frac{3 \pi}{16}\)
03

Use Addition of Angles

Adding the angles in \(\cos(A + B)\) gives the expression \(\cos(\frac{\pi}{16} + \frac{3 \pi}{16})\)
04

Simplify Expression

The addition inside the cosine gives \(\cos(\frac{4\pi}{16})\), which simplifies to \(\cos(\frac{\pi}{4})\).

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