Chapter 9.7, Problem 33E

To determine

To find: If both answers are equivalent or not.

Explanation of Solution

Given information:

Given −

  $\sec \left(-\theta \right)=\sec \theta \text{ or }\sec \left(-\theta \right)=-\sec \theta$ .

Calculation:

A trigonometric equation is true for only certain values of variables while a trigonometric identity is true for all values of variables. In a trigonometric equation expressions on both side of equal sign are equal while in case of trigonometric identity expression on both side of equal sign are equivalent. While verifying trigonometric identity, we do not assume that both sides are equal. Therefore, any properties of equality cannot be used for verifying.

  $\begin{array}{l}\text{Now let us verify the identity }\sec x\tan x-\sin x=\sin x{\tan }^{2}x,\\ \text{LHS}=\sec x\tan x-\sin x\\ =\frac{1}{\cos x}\cdot \frac{\sin x}{\cos x}-\sin x\\ =\frac{\sin x-\sin x{\cos }^{2}x}{{\cos }^{2}x}\\ =\frac{\sin x\left(1-{\cos }^{2}x\right)}{{\cos }^{2}x}\\ =\frac{\sin x{\sin }^{2}x}{{\cos }^{2}x}\left\{\text{Since, }1-{\cos }^{2}x={\sin }^{2}x\right\}\\ =\sin x\cdot \frac{{\sin }^{2}x}{{\cos }^{2}x}\\ =\sin x{\tan }^{2}x\\ =\text{RHS}\end{array}$

Hence, it can be said that the both answers are equivalent.