Chapter 7.1, Problem 15E

To determine

To simplify the given expression.

Answer to Problem 15E

  $\frac{1+\cos y}{1+\sec y}=\cos y$

Explanation of Solution

Given:

An expression is given as

  $\frac{1+\cos y}{1+\sec y}$

Concept Used:

The trigonometric identities:

  $\sec x=\frac{1}{\cos x}$

Calculation:

In order to simplify the expression:

  $\frac{1+\cos y}{1+\sec y}$

Now, simplify the given expression as shown:

  $\begin{array}{c}\frac{1+\cos y}{1+\sec y}=\frac{1+\cos y}{1+\frac{1}{\cos y}}\\ =\frac{1+\cos y}{\frac{1+\cos y}{\cos y}}\\ =\left(1+\cos y\right)\times \frac{\cos y}{1+\cos y}\\ =\cos y\end{array}$

Thus, the simplified form of the expression is

  $\frac{1+\cos y}{1+\sec y}=\cos y$