Chapter 5, Problem 18RE

To determine

To simplify: the expression and check result numerically.

Answer to Problem 18RE

  $\frac{{\sec }^2\left(-\theta \right)}{{\csc }^2\theta }={\tan }^2\theta$

Explanation of Solution

Given information:

Given expression

  $\frac{{\sec }^2\left(-\theta \right)}{{\csc }^2\theta }$ .

Calculation:

Let us consider the following expression, $\frac{{\sec }^2\left(-\theta \right)}{{\csc }^2\theta }$

The given trigonometric expression requires the use of the identities given by equation (1), (2), (3) and (4). Equation (1) gives that is known as the even-odd identity of cosine of $x$ .

  $\begin{array}{l}\cos \left(-x\right)=\cos \left(x\right)\mathrm{................}(1)\\ \sec \left(x\right)=\frac{1}{\cos x}\mathrm{...............}(2)\\ \csc x=\frac{1}{\sin x}\mathrm{................}(3)\\ \tan x=\frac{\sin x}{\cos x}\mathrm{..................}(4)\end{array}$

The identities given in equations (2) and (3) are used in simplification of the given trigonometric expression in the following way.

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

Now the use of the even-odd identity of cosine function given by equation (1) is used to further simplify the expression.

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

The identity given by equation (4) is to be used for final simplified form of the expression.

  ${\left(\tan \theta \right)}^2={\tan }^2\theta$

Hence, $\frac{{\sec }^2\left(-\theta \right)}{{\csc }^2\theta }={\tan }^2\theta$ .

Now, check the result graphically as shown:

Table of values for $\frac{{\sec }^2\left(-\theta \right)}{{\csc }^2\theta }$ is shown:

  

Graph of $\frac{{\sec }^2\left(-\theta \right)}{{\csc }^2\theta }$ is constructed as

  

Table of values for ${\tan }^2\theta$ is shown:

  

Graph of ${\tan }^2\theta$ is constructed as shown:

  

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