Chapter 5.2, Problem 21E

To determine

To verify the identity:

  $\frac{1+\sin \theta }{\cos \theta }+\frac{\cos \theta }{1+\sin \theta }=2\sec \theta$

And also verify this using table feature of graphing utility.

Explanation of Solution

Given:

  $\frac{1+\sin \theta }{\cos \theta }+\frac{\cos \theta }{1+\sin \theta }=2\sec \theta$

Concept Used:

The trigonometric identities:

 • $\sec x=\frac{1}{\cos x}$

 • ${\sin }^{2}x+{\cos }^{2}x=1$

Calculation:

In order to verify the identity:

  $\frac{1+\sin \theta }{\cos \theta }+\frac{\cos \theta }{1+\sin \theta }=2\sec \theta$

Take left hand of above identity and simplify it using the trigonometric identities.

As,

  $\begin{array}{c}\frac{1+\sin \theta }{\cos \theta }+\frac{\cos \theta }{1+\sin \theta }=\frac{{\left(1+\sin \theta \right)}^2+{\left(\cos \theta \right)}^2}{\cos \theta \left(1+\sin \theta \right)}\\ =\frac{1+{\sin }^2\theta +2\sin \theta +{\cos }^2\theta }{\cos \theta \left(1+\sin \theta \right)}\\ =\frac{1+{\sin }^2\theta +{\cos }^2\theta +2\sin \theta }{\cos \theta \left(1+\sin \theta \right)}\\ =\frac{2+2\sin \theta }{\cos \theta \left(1+\sin \theta \right)}\\ =\frac{2\left(1+\sin \theta \right)}{\cos \theta \left(1+\sin \theta \right)}\\ =2\sec \theta \\ =\text{Right hand}\end{array}$

Thus, identity is verified, i.e.

  $\frac{1+\sin \theta }{\cos \theta }+\frac{\cos \theta }{1+\sin \theta }=2\sec \theta$

Table of values for $\frac{1+\sin \theta }{\cos \theta }+\frac{\cos \theta }{1+\sin \theta }$ is shown:

  

Graph of $\frac{1+\sin \theta }{\cos \theta }+\frac{\cos \theta }{1+\sin \theta }$ is constructed as

  

Table of values for $2\sec \theta$ is shown:

  

Graph of $2\sec \theta$ is constructed as shown:

  

Since, the graph of left hand side and right hand side of the identity coincides, thus the identity is verified by graph too.