Chapter 7.1, Problem 51E

To determine

To simplify given expression.

Answer to Problem 51E

  $1+{\cot }^2\theta -{\cos }^2\theta -{\cos }^2\theta {\cot }^2\theta =1$

Explanation of Solution

Given information :

  $1+{\cot }^2\theta -{\cos }^2\theta -{\cos }^2\theta {\cot }^2\theta$

We shall use the following identites,

  $\begin{array}{l}1+{\cot }^2\theta ={\csc }^2\theta \\ {\sin }^2\theta +{\cos }^2\theta =1\end{array}$

Hence we get,

  $\begin{array}{c}1+{\cot }^2\theta -{\cos }^2\theta -{\cos }^2\theta {\cot }^2\theta =1+{\cot }^2\theta -{\cos }^2\theta \left(1+{\cot }^2\theta \right)\\ ={\csc }^2\theta -{\cos }^2\theta \left({\csc }^2\theta \right)\\ ={\csc }^2\theta \left(1-{\cos }^2\theta \right)\\ ={\csc }^2\theta {\sin }^2\theta \\ =\frac{{\sin }^2\theta }{{\sin }^2\theta }\\ =1\end{array}$