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Chapter 5: Problem 10

Verify each identity. $$\frac{\cos (\alpha-\beta)}{\sin \alpha \sin \beta}=\cot \alpha \cot \beta+1$$

Short Answer

Expert verified
The identity holds as the left side simplifies exactly to \(\cot \alpha \cot \beta+1\) which is the right side.

Step by step solution

01

Express \(\cos (\alpha-\beta)\)

The formula for \(\cos(\alpha-\beta)\) can be expressed as \(\cos \alpha \cos \beta + \sin \alpha \sin \beta\). Substituting this into the original equation, we will have \[\frac{\cos \alpha \cos \beta + \sin \alpha \sin \beta}{\sin \alpha \sin \beta}\].
02

Divide the expression

Now, divide each term in the numerator by \(\sin \alpha \sin \beta\) to separate the fraction. The expression then becomes \[\frac{\cos \alpha \cos \beta}{\sin \alpha \sin \beta} + \frac{\sin \alpha \sin \beta}{\sin \alpha \sin \beta}\].
03

Express into Cotangent form and simplify

Express \(\frac{\cos \alpha }{\sin \alpha }\) and \(\frac{\cos \beta}{\sin \beta}\) as \(\cot \alpha\) and \(\cot \beta\) respectively and simplify: \[\cot \alpha \cot \beta+ 1\].

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