Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1

Apply the product rule to $\frac{1}{\cos (x - 1) \sin (x)}$ .

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

Step 5.2

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

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

Step 6

One to any power is one.

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

Step 7

Multiply by $1$ .

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

Step 8

Separate fractions.

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

Step 9

Convert from $\frac{1}{\cos^{2} (x - 1)}$ to $\sec^{2} (x - 1)$ .

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

Step 10

Write the expression using exponents.

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Step 10.1

Multiply $\sin^{2} (x)$ by $1$ .

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

Step 10.2

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

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

Step 11

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

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

Step 12

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

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