Chapter 5, Problem 16RE

To determine

To simplify: the expression, check result numerically.

Answer to Problem 16RE

  $\frac{{\sin }^2\theta +{\cos }^2\theta }{\sin \theta }=\csc \theta$

Explanation of Solution

Given information:

Given expression

  $\frac{{\sin }^2\theta +{\cos }^2\theta }{\sin \theta }$ .

Calculation:

Let us consider the following expression, $\frac{{\sin }^2\theta +{\cos }^2\theta }{\sin \theta }$

The given trigonometric expression can be simplified using the identities given by equation (1) and (2)

  $\begin{array}{l}{\sin }^{2}x+{\cos }^{2}x=\mathrm{1..............}(1)\\ \csc x=\frac{1}{\sin x}\mathrm{....................}(2)\end{array}$

The identities given by equations (1) and (2) are used for simplification of given expression in the following way.

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

Now, check the result graphically as shown:

Table of values for $\frac{{\sin }^2\theta +{\cos }^2\theta }{\sin \theta }$ is shown:

  

Graph of $\frac{{\sin }^2\theta +{\cos }^2\theta }{\sin \theta }$ is constructed as

  

Table of values for $\csc \theta$ is shown:

  

Graph of $\csc \theta$ is constructed as shown:

  

Both graphs are same and the functional values are also same.