Chapter 7, Problem 17RE

To determine

To prove: The identity $1+\tan x\tan \frac{x}{2}=\sec x$ .

Explanation of Solution

Given information:

The given identity is $1+\tan x\tan \frac{x}{2}=\sec x$ .

Formula used:

Identity:

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

Reciprocal identity:

  $\sec x=\frac{1}{\cos x}$

Proof:

To prove the identity, take left side of the identity and simplify it.

  $\begin{array}{c}1+\tan x\tan \frac{x}{2}=1+\left(\frac{\sin x}{\cos x}\right)\left(\frac{1-\cos x}{\sin x}\right)\\ =1+\frac{1-\cos x}{\cos x}\\ =1+\frac{1}{\cos x}-1\\ =\sec x\end{array}$

From the above, it is clear that the left hand side of the identity is equal to the right side of the identity.

Hence, proved.