Chapter 5.1, Problem 49E

Solution

To find: The algebraic expression with single trigonometric function for $\frac{{\sin }^{2}x}{1+\cos x}$ .

The algebraic expression with single trigonometric function for $\frac{{\sin }^{2}x}{1+\cos x}$ is $\left(1-\cos x\right)$ .

Given information:

The given expression is $\frac{{\sin }^{2}x}{1+\cos x}$ .

Formula used:

The Pythagorean identity is given by,

  $\begin{array}{c}{\cos }^{2}x+{\sin }^{2}x=1\\ {\sin }^{2}x=1-{\cos }^{2}x\end{array}$

The formula for the difference of squares of two terms is given by,

  $a^2-b^2=\left(a+b\right)\left(a-b\right)$

Calculation:

Consider the given expression,

  $\Rightarrow \frac{{\sin }^{2}x}{1+\cos x}$

Use the Pythagorean identity and substitute $1-{\cos }^{2}x$ for ${\sin }^{2}x$ in the expression.

  $\Rightarrow \frac{{\sin }^{2}x}{1+\cos x}=\frac{1-{\cos }^{2}x}{1+\cos x}$

Use the identity for the difference of squares of two terms to simplify the numerator of the expression.

  $\Rightarrow \frac{\left(1-\cos x\right)\left(1+\cos x\right)}{1+\cos x}$

Cancel the similar terms, $\left(1+\cos x\right)$ .

  $\Rightarrow \left(1-\cos x\right)$

Thus, the algebraic expression with single trigonometric function for $\frac{{\sin }^{2}x}{1+\cos x}$ is $\left(1-\cos x\right)$ .