Chapter 7.2, Problem 15E

To determine

To verify: that the given equation is an identity.

Answer to Problem 15E

See explanation

Explanation of Solution

Given information:

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

Calculation:

Simplify the right side of the equation into left side as follows:

  $\begin{array}{c}\sec x-\tan x=\frac{1-\sin x}{\cos x}\end{array}$

  $\begin{array}{c}=\frac{1}{\cos x}-\frac{\sin x}{\cos x}\end{array}$

  $\begin{array}{c}=\sec x-\tan x\left[\begin{array}{l}\text{Use quotient identity for }\tan x\\ \text{Use reciprocal identity of }\sec x\end{array}\right]\end{array}$

  $\begin{array}{c}LHS=RHS\end{array}$

Since the right side of the equation is transformed into left hand side. So, the equation is an identity.