Chapter 7.2, Problem 13E

To determine

To verify: that the given equation is an identity.

Answer to Problem 13E

See explanation

Explanation of Solution

Given information:

  $\tan A=\frac{\sec A}{\csc A}$

Calculation:

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

  $\begin{array}{c}\tan A=\frac{\sec A}{\csc A}\end{array}$

  $\begin{array}{c}=\frac{\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\cos A$}\right.\right)}{\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sin A$}\right.\right)}\left[\text{Use reciprocal identities}\right]\end{array}$

  $\begin{array}{c}=\frac{\sin A}{\cos A}\end{array}$

  $\begin{array}{c}=\tan A\left[\text{Use quotient identity: }\tan x=\frac{\sin x}{\cos x}\right]\end{array}$

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

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