Chapter 7.1, Problem 50E

To determine

To simplify given expression.

Answer to Problem 50E

  $\left(1+\cos \theta \right)\left(\csc \theta -\cot \theta \right)=\sin \theta$

Explanation of Solution

Given information :

  $\left(1+\cos \theta \right)\left(\csc \theta -\cot \theta \right)$

we know that,

  $\begin{array}{l}\csc \theta =\frac{1}{\sin \theta }\\ \cot \theta =\frac{\cos \theta }{\sin \theta }\end{array}$

Hence we get,

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