Trigonometry Examples

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

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

Step 1

Start on the left side.

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

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

Step 2

Convert to sines and cosines.

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Step 2.1

Apply the reciprocal identity to

csc (x)

$\csc (x)$ .

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

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

Step 2.2

Write

cot (x)

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

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

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

Step 2.3

Apply the product rule to

\frac{1}{sin (x)}

$\frac{1}{\sin (x)}$ .

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

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

Step 2.4

Apply the product rule to

\frac{cos (x)}{sin (x)}

$\frac{\cos (x)}{\sin (x)}$ .

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

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

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

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

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}

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

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}

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

Step 3.3

Simplify the denominator.

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Step 3.3.1

Rewrite

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

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

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

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

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

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

Step 3.3.2

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = \frac{cos (x)}{sin (x)}

$a = \frac{\cos (x)}{\sin (x)}$ and

b = 1

$b = 1$ .

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

$1$ 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)}

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

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)}

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

Step 3.3.6

To write

- 1

$- 1$ as a fraction with a common denominator, multiply by

\frac{sin (x)}{sin (x)}

$\frac{\sin (x)}{\sin (x)}$ .

\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)})}

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

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