Chapter 7, Problem 57RE

To determine

To establish:the given identity.

Answer to Problem 57RE

The identity is established.

Explanation of Solution

Given:

The given identity is $\frac{1-2{\sin }^2\theta }{\sin \theta \cos \theta }=\cot \theta -tan\theta$ .

Calculation:

As the given identity is $\frac{1-2{\sin }^2\theta }{\sin \theta \cos \theta }=\cot \theta -tan\theta$ .

Now, let us check RHS:

  $\begin{array}{l}RHS=\cot \theta -tan\theta \\ =\frac{\cos \theta }{sin\theta }-\frac{\sin \theta }{\cos \theta }\{\text{as }\cot \theta =\frac{\cos \theta }{\sin \theta },tan\theta =\frac{sin\theta }{\cos \theta }\}\\ =\frac{{\cos }^2\theta -si{n}^2\theta }{sin\theta \cos \theta }\{\text{as }{\cos }^2\theta =\left(1-{\sin }^2\theta \right)\}\\ =\frac{\left(1-{\sin }^2\theta \right)-si{n}^2\theta }{sin\theta \cos \theta }\\ =\frac{1-2{\sin }^2\theta }{\sin \theta \cos \theta }=LHS\end{array}$

Thus, $\frac{1-2{\sin }^2\theta }{\sin \theta \cos \theta }=\cot \theta -tan\theta$

Hence, the identity is established.