Chapter 7, Problem 62RE

To determine

To establish:the given identity.

Answer to Problem 62RE

The identity is established.

Explanation of Solution

Given:

The given identity is $\frac{\cos \left(\alpha +\beta \right)}{\sin \alpha \cos \beta }=\cot \alpha -\tan \beta$ .

Calculation:

As the given identity is $\frac{\cos \left(\alpha +\beta \right)}{\sin \alpha \cos \beta }=\cot \alpha -\tan \beta$ .

Now, let us check LHS:

  $\begin{array}{l}LHS=\frac{\cos \left(\alpha +\beta \right)}{\sin \alpha \cos \beta }\\ \left\{\cos \left(A+B\right)=\cos A\cos B-\sin A\sin B\right\}\\ =\frac{\cos \alpha \cos \beta -\sin \alpha \sin \beta }{\sin \alpha \cos \beta }\\ =\frac{\cos \alpha \cos \beta }{\sin \alpha \sin \beta }-\frac{\sin \alpha \sin \beta }{\sin \alpha \sin \beta }\\ =\cot \alpha -\tan \beta \\ =RHS\end{array}$

Thus, $\frac{sin\left(\alpha -\beta \right)}{sin\alpha \cos \beta }=1-\cot \alpha \tan \beta$

Hence, the identity is established.