Trigonometry Examples

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

Step 1

Apply pythagorean identity.

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

Step 2

Apply pythagorean identity.

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

Step 3

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

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

Step 4

Simplify the numerator.

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Step 4.1

Since $\cot (- x)$ is an odd function, rewrite $\cot (- x)$ as $- \cot (x)$ .

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

Step 4.2

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

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

Step 5

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

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

Step 6

Dividing two negative values results in a positive value.

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

Step 7

Multiply the numerator by the reciprocal of the denominator.

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Add $1$ and $1$ .

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

Step 9

Apply the product rule to $\frac{\cos (x)}{\sin^{2} (x)}$ .

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

Step 10

Multiply the exponents in ${(\sin^{2} (x))}^{2}$ .

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Step 10.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 10.2

Multiply $2$ by $2$ .

$\frac{\cos^{2} (x)}{\sin^{4} (x)}$

Step 11

Multiply by $1$ .

$\frac{\cos^{2} (x) \cdot 1}{\sin^{4} (x)}$

Step 12

Factor $\sin^{2} (x)$ out of $\sin^{4} (x)$ .

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

Step 13

Separate fractions.

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

Step 14

Convert from $\frac{\cos^{2} (x)}{\sin^{2} (x)}$ to $\cot^{2} (x)$ .

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

Step 15

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

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

Step 16

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

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

Step 17

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

$\csc^{2} (x) \cot^{2} (x)$