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Chapter 5: Problem 122

Verify the identity. $$\frac{\sin x \pm \sin y}{\cos x+\cos y}=\tan \frac{x \pm y}{2}$$

Short Answer

Expert verified
After applying the addition and subtraction formulas of sine and cosine functions, and simplifying the resulting expressions, the given identity \(\frac{\sin x \pm \sin y}{\cos x+\cos y}=\tan \frac{x \pm y}{2}\) has been verified as true.

Step by step solution

01

Apply addition or subtraction rule for sine

To begin with, use the addition or subtraction rule for sine: \(\sin (a+b)=\sin a \cos b + \cos a \sin b\) or \(\sin (a-b)=\sin a \cos b - \cos a \sin b\). Apply this to \(\sin x \pm \sin y\), and get \(\sin x \cos y \pm \cos x \sin y\)
02

Apply cosine addition formula

Next, apply the cosine addition formula: \(\cos (a+b)=\cos a \cos b - \sin a \sin b\) or \( \cos (a-b)=\cos a \cos b + \sin a \sin b\) to \( \cos x+ \cos y\), to get \(\cos x \cos y -\sin x \sin y\) or \(\cos x \cos y + \sin x \sin y\).\
03

Simplify the expression

Then, simplify fraction with the term obtained from step 1 as numerator and the term from step 2 as denominator, we have the simplified version: \( \frac{\sin x \cos y \pm \cos x \sin y}{\cos x \cos y \mp \sin x \sin y}\). Now, see that expression is equal to \( \tan \frac{x \pm y}{2}\) as needed.

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