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Chapter 6: Problem 91

Use the sum-to-product formulas to find the exact value of the expression. $$\cos \frac{3 \pi}{4}-\cos \frac{\pi}{4}$$

Short Answer

Expert verified
-\sqrt{2}

Step by step solution

01

Recognize the form

The provided expression corresponds to \( -2 \sin \frac{a+b}{2} \sin \frac{a-b}{2} \) where \( a = \frac{3\pi}{4} \) and \( b = \frac{\pi}{4} \)
02

Plug the values into the formula

On using these values in the formula, we get \( -2 \sin \left( \frac{\frac{3\pi}{4} + \frac{\pi}{4}}{2}\right) \sin \left( \frac{\frac{3\pi}{4} - \frac{\pi}{4}}{2}\right) \)
03

Simplify the expression

Simplifying the expression, we get \( -2 \sin \left( \frac{\pi}{2}\right) \sin \left( \frac{\pi}{4}\right) \)
04

Calculate the sine values

Let's calculate the sine values of these two angles. \( \sin \frac{\pi}{2} = 1 \) and \( \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \)
05

Complete the Calculation

Plugging these sine values back into the expression, we get \( -2 \times 1 \times \frac{1}{\sqrt{2}} = -\sqrt{2} \)

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