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Chapter 5: Problem 69

Prove the identity. $$\sin (x+y)+\sin (x-y)=2 \sin x \cos y$$

Short Answer

Expert verified
The identity \(\sin(x+y) + \sin(x-y) = 2\sin(x)\cos(y)\) is proven by using the trigonometric addition formulas, substituting them into the left side of the equation, and simplifying the resulting expression. In the end, the left side of the given identity equation matches the right side, which verifies the identity.

Step by step solution

01

Write down the trigonometric addition formulas

The trigonometric addition formulas are: \(\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b)\), and \(\sin(a-b) = \sin(a)\cos(b) - \cos(a)\sin(b)\).
02

Subsitute the formulas into the left side of the identity

Replace \(\sin(x+y)\) with \(\sin(x)\cos(y) + \cos(x)\sin(y)\) and replace \(\sin(x-y)\) with \(\sin(x)\cos(y) - \cos(x)\sin(y)\). Plugging these two formulas into the original equation results in: \[\sin(x)\cos(y) + \cos(x)\sin(y) + \sin(x)\cos(y) - \cos(x)\sin(y)\].
03

Simplify the obtained expression

When simplified, the equation becomes: \[2\sin(x)\cos(y)\].

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