/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 Verify each identity. $$\cos (... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 40

Verify each identity. $$\cos (\alpha+\beta)+\cos (\alpha-\beta)=2 \cos \alpha \cos \beta$$

Short Answer

Expert verified
The left side of given identity, when applied the cosine addition and subtraction identities and simplified, matches exactly with the right side. Hence, the given identity \( \cos (\alpha+\beta)+\cos (\alpha-\beta)=2 \cos \alpha \cos \beta \) has been verified.

Step by step solution

01

Apply the cosine addition and subtraction identities

To begin with, the given expression is: \(\cos (\alpha+\beta)+\cos (\alpha-\beta)\) and we need to transform it into the form: \(2 \cos \alpha \cos \beta\).Since the Cosine of the sum of two angles identity is \( \cos (\alpha+\beta)= \cos \alpha \cos \beta - \sin \alpha \sin \beta\) and the Cosine of the difference of two angles identity is \(\cos (\alpha-\beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\), we can write the original expression as:\(\cos (\alpha+\beta)+\cos (\alpha-\beta)= \cos \alpha \cos \beta - \sin \alpha \sin \beta + \cos \alpha \cos \beta + \sin \alpha \sin \beta\)
02

Simplify the expression

Adding the similar terms from the right side, it leads to: \( 2\cos \alpha \cos \beta\). Thus simplifying the given identity now leads to the desired form.

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Most popular questions from this chapter

Find the exact value of each expression. Do not use a calculator. $$\cos \left[\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)-\sin ^{-1}\left(-\frac{1}{2}\right)\right]$$

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