Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

Apply the reciprocal identity to $\csc (x)$ .

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

Step 2.2

Write $\cot (x)$ in sines and cosines using the quotient identity.

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Simplify.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

One to any power is one.

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

Step 3.3

Simplify the denominator.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{\cos (x)}{\sin (x)}$ and $b = 1$ .

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

Step 3.3.4

Write $1$ as a fraction with a common denominator.

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

Step 3.3.5

Combine the numerators over the common denominator.

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

Step 3.3.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{\sin (x)}{\sin (x)}$ .

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

Step 3.3.7

Combine $- 1$ and $\frac{\sin (x)}{\sin (x)}$ .

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

Step 3.3.8

Combine the numerators over the common denominator.

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

Step 3.4

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

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

Step 3.5

Simplify the denominator.

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.5.4

Add $1$ and $1$ .

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

Step 3.6

Combine.

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

Step 3.7

Multiply $1$ by $1$ .

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

Step 3.8

Combine ${\sin (x)}^{2}$ and $\frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}{{\sin (x)}^{2}}$ .

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

Step 3.9

Reduce the expression by cancelling the common factors.

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

Step 4

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

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

Step 5

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{\csc^{2} (x)}{\cot^{2} (x) - 1} = \frac{\sec^{2} (x)}{1 - \tan^{2} (x)}$ is an identity