Precalculus Examples

$\frac{\csc^{2} (x - 1)}{\csc^{2} (x)}$

Step 1

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

${(\frac{\csc (x - 1)}{\csc (x)})}^{2}$

Step 2

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

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

Step 3

Rewrite $\csc (x - 1)$ in terms of sines and cosines.

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

Step 4

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

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

Step 5

Write $\sin (x)$ as a fraction with denominator $1$ .

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

Step 6

Simplify.

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Step 6.1

Divide $\sin (x)$ by $1$ .

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

Step 6.2

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

${(\csc (x - 1) \sin (x))}^{2}$

Step 7

Apply the product rule to $\csc (x - 1) \sin (x)$ .

$\csc^{2} (x - 1) \sin^{2} (x)$