Chapter 7.1, Problem 37E

To determine

To verify the identity:

  $\tan \theta +\cot \theta =\sec \theta \csc \theta$

Explanation of Solution

Given:

An identity is given as

  $\tan \theta +\cot \theta =\sec \theta \csc \theta$

Concept Used:

The trigonometric identities:

  $\begin{array}{l}\tan x=\frac{\sin x}{\cos x}\\ {\sin }^{2}x+{\cos }^{2}x=1\\ \cot x=\frac{\cos x}{\sin x}\\ \csc x=\frac{1}{\sin x}\\ \sec x=\frac{1}{\cos x}\end{array}$

Calculation:

In order to verify the identity:

  $\tan \theta +\cot \theta =\sec \theta \csc \theta$

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

As,

  $\begin{array}{c}\tan \theta +\cot \theta =\frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta }\\ =\frac{\sin \theta \times \sin \theta +\cos \theta \times \cos \theta }{\cos \theta \sin \theta }\\ =\frac{{\sin }^2\theta +{\cos }^2\theta }{\cos \theta \sin \theta }\\ =\frac{1}{\cos \theta \sin \theta }\\ =\frac{1}{\cos \theta }\times \frac{1}{\sin \theta }\\ =\sec \theta \csc \theta \\ =\text{Right hand}\end{array}$

Thus, identity is verified, i.e.

  $\tan \theta +\cot \theta =\sec \theta \csc \theta$