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Chapter 5: Problem 38

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 simplified form of the given expression is \(\cos(\frac{\pi}{8})\).

Step by step solution

01

Identify the Associative Form

First, identify the formula pattern within the given expression. It relates to the cosine of the difference of two angles formula, which is \(\cos(a-b)=\cos a\cos b +\sin a\sin b\), except there is a negative sign between the two terms, hence it is subtraction instead of addition: \(\cos a\cos b -\sin a\sin b\). Here, \(\cos \frac{\pi}{16}\cos \frac{3 \pi}{16}-\sin \frac{\pi}{16} \sin \frac{3\pi}{16}\) is in the pattern of \(\cos(a-b)\). So, \(a=\frac{\pi}{16}\) and \(b=\frac{3\pi}{16}\)
02

Use the Cosine Angle Subtraction Formula

Next, apply the reverse of the cosine angle difference formula to the expression, with \(a=\frac{\pi}{16}\) and \(b=\frac{3\pi}{16}\). This turns the expression into \(\cos(\frac{\pi}{16}-\frac{3\pi}{16})\)
03

Simplify the expression

Then, simplify the expression \(\cos(\frac{\pi}{16}-\frac{3\pi}{16})\) to \(\cos(-\frac{2\pi}{16})\). The negative sign can be ignored in the case of cosine as it is an even function, which leads to \(\cos(\frac{2\pi}{16})\).
04

Further Simplification

Further reduce the expression \(\cos(\frac{2\pi}{16})\) to \(\cos(\frac{\pi}{8})\) by simplifying the fraction in the brackets.

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