Trigonometry Examples

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

Step 1

Apply pythagorean identity.

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

Step 2

Apply pythagorean identity.

$\frac{\tan^{2} (x)}{\cot^{2} (x)}$

Step 3

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

${(\frac{\tan (x)}{\cot (x)})}^{2}$

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

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

Step 7.2

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

${(\tan (x) \tan (x))}^{2}$

Step 7.3

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

${(\tan^{1} (x) \tan (x))}^{2}$

Step 7.4

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

${(\tan^{1} (x) \tan^{1} (x))}^{2}$

Step 7.5

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

${({\tan (x)}^{1 + 1})}^{2}$

Step 7.6

Add $1$ and $1$ .

${(\tan^{2} (x))}^{2}$

Step 8

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

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Step 8.1

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

${\tan (x)}^{2 \cdot 2}$

Step 8.2

Multiply $2$ by $2$ .

$\tan^{4} (x)$