Precalculus Examples

$\frac{\csc (x)}{\sin (x)}$

Step 1

Rewrite $\csc (x)$ in terms of sines and cosines.

$\frac{\frac{1}{\sin (x)}}{\sin (x)}$

Step 2

Rewrite $\frac{\frac{1}{\sin (x)}}{\sin (x)}$ as a product.

$\frac{1}{\sin (x)} \cdot \frac{1}{\sin (x)}$

Step 3

Multiply $\frac{1}{\sin (x)}$ by $\frac{1}{\sin (x)}$ .

$\frac{1}{\sin (x) \sin (x)}$

Step 4

Simplify the denominator.

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Step 4.1

Raise $\sin (x)$ to the power of $1$ .

$\frac{1}{\sin^{1} (x) \sin (x)}$

Step 4.2

Raise $\sin (x)$ to the power of $1$ .

$\frac{1}{\sin^{1} (x) \sin^{1} (x)}$

Step 4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{{\sin (x)}^{1 + 1}}$

Step 4.4

Add $1$ and $1$ .

$\frac{1}{\sin^{2} (x)}$

Step 5

Simplify with factoring out.

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Step 5.1

Rewrite $1$ as $1^{2}$ .

$\frac{1^{2}}{\sin^{2} (x)}$

Step 5.2

Rewrite $\frac{1^{2}}{\sin^{2} (x)}$ as ${(\frac{1}{\sin (x)})}^{2}$ .

${(\frac{1}{\sin (x)})}^{2}$

Step 6

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\csc^{2} (x)$