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Chapter 5: Problem 40

Verify the identity. $$\frac{\cos x-\cos y}{\sin x+\sin y}+\frac{\sin x-\sin y}{\cos x+\cos y}=0$$

Short Answer

Expert verified
The verification of the identity results in a zero in the numerator of the complex fraction, confirming the identity.

Step by step solution

01

Simplify Denominator

Common denominator for the two fractions on the left side of equality is \((\sin x+\sin y)(\cos x+\cos y)\). Rewrite the fractions with common denominator so that they can be combined.
02

Apply Trigonometric Formulas

Now apply the formula for the difference of two squares to the denominator, \((a+b)(a-b)=a^2-b^2\), in our case we will have \(\sin^2x-\cos^2y\) in the denominator.
03

Combine Numerator Terms

Combine the numerators under the common denominator. Apply the formula for the sum and difference of sine and cosine, \(\sin a\cos b-\cos a \sin b = \sin(a-b)\), and \(\cos a\cos b+\sin a\sin b = \cos(a-b)\). In the equation, these will become \(\cos(x-y)\) and \(\sin(x-y)\).
04

Simplify Expression

Subtracting the two, we will have the numerator equaling to zero, which verifies the given identity.

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