Chapter 5.2, Problem 53E

Solution

To find: An equivalent expression from the given options.

The given expression is equivalent to the expression in option (d).

Given information:

The given expression is $\left(1+\sec x\right)\left(1-\cos x\right)$ .

Formula used:

The trigonometric identities are $\sec x\cos x=1$, $1-{\cos }^{2}x={\sin }^{2}x$ and $\tan x=\frac{\sin x}{\cos x}$ .

The distributive property is $\left(a+b\right)\left(c+d\right)=ac+ad+bc+bd$

Calculation:

Rewrite the given expression by using the distributive propertyand trigonometric identities as follows.

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

Simplify the obtained expression using trigonometric identity.

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

This implies that the given expression is equivalent to $\tan x\sin x$ . Therefore, the correct option is (d).