Chapter 6, Problem 11CLT

To determine

To find: The simplified form of the give equation.

Answer to Problem 11CLT

Thesimplified form of the given trigonometric equation is $2\tan \theta$ .

Explanation of Solution

Given:

The given equation is,

  $\frac{\sin \theta -1}{\cos \theta }-\frac{\cos \theta }{\sin \theta -1}$

Calculation:

Consider the given equation is,

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

Solve further as,

  $\frac{\sin \theta -1}{\cos \theta }-\frac{\cos \theta }{\sin \theta -1}=2\tan \theta$