Question

perform the indicated operation and simplify the result so that there are no quotients.

sin x + \frac{cos^{2}x}{sin x}

sin x + \frac{cos^{2}x}{sin x} = \square (simplify your answer.)

Explanation:

Step1: Find a common denominator

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

Step2: Combine the fractions

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

Step3: Use the Pythagorean identity

Since $\sin^{2}x+\cos^{2}x = 1$, the expression becomes $\frac{1}{\sin x}$.

Step4: Simplify using reciprocal identity

$\frac{1}{\sin x}=\csc x$

Answer:

$\csc x$