Chapter 5.1, Problem 67E

To determine

To express: The given trigonometric expression in non-fractional form.

Answer to Problem 67E

The expression $\frac{{\sin }^{2}y}{1-\cos y}$ is equal to $(1+\cos y)$ .

Explanation of Solution

Given information:

The given trigonometric expression:

  $\frac{{\sin }^{2}y}{1-\cos y}$

Formula used:

Pythagorean identity:

  ${\sin }^{2}y=1-{\cos }^{2}y=(1+\cos y)(1-\cos y)$

Calculation:

As per problem,

The given expression:

  $\frac{{\sin }^{2}y}{1-\cos y}$

Rewrite the numerator using the Pythagorean identity: ${\sin }^{2}y=1-{\cos }^{2}y=(1+\cos y)(1-\cos y)$

  $\begin{array}{l}\frac{{\sin }^{2}y}{1-\cos y}=\frac{(1+\cos y)(1-\cos y)}{(1-\cos y)}\text{ [pythagorean identity] }\\ \text{ =}\frac{(1+\cos y)\cancel{(1-\cos y)}}{\cancel{(1-\cos y)}}\text{ [cancel out common factors]}\\ \text{ =1+}\cos y\text{ [simplify]}\end{array}$

Hence, the expression $\frac{{\sin }^{2}y}{1-\cos y}$ is equal to $(1+\cos y)$ .