Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

Write

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

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

Step 2.2

Apply the reciprocal identity to

$\csc (x)$ .

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

Step 2.3

Simplify.

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Step 2.3.1

Rewrite

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

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

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

Step 2.3.2

Expand

$(\frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)}) (\frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)})$ using the FOIL Method.

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Step 2.3.2.1

Apply the distributive property.

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

Step 2.3.2.2

Apply the distributive property.

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

Step 2.3.2.3

Apply the distributive property.

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

Step 2.3.3

Simplify and combine like terms.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Multiply

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

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Step 2.3.3.1.1.1

Multiply

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

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

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

Step 2.3.3.1.1.2

Raise

$\cos (x)$ to the power of

$1$ .

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

Step 2.3.3.1.1.3

Raise

$\cos (x)$ to the power of

$1$ .

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

Step 2.3.3.1.1.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3.3.1.1.5

Add

$1$ and

$1$ .

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

Step 2.3.3.1.1.6

Raise

$\sin (x)$ to the power of

$1$ .

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

Step 2.3.3.1.1.7

Raise

$\sin (x)$ to the power of

$1$ .

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

Step 2.3.3.1.1.8

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3.3.1.1.9

Add

$1$ and

$1$ .

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

Step 2.3.3.1.2

Multiply

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

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Step 2.3.3.1.2.1

Multiply

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

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

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

Step 2.3.3.1.2.2

Raise

$\sin (x)$ to the power of

$1$ .

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

Step 2.3.3.1.2.3

Raise

$\sin (x)$ to the power of

$1$ .

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

Step 2.3.3.1.2.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3.3.1.2.5

Add

$1$ and

$1$ .

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

Step 2.3.3.1.3

Multiply

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

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Step 2.3.3.1.3.1

Multiply

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

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

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

Step 2.3.3.1.3.2

Raise

$\sin (x)$ to the power of

$1$ .

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

Step 2.3.3.1.3.3

Raise

$\sin (x)$ to the power of

$1$ .

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

Step 2.3.3.1.3.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3.3.1.3.5

Add

$1$ and

$1$ .

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

Step 2.3.3.1.4

Multiply

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

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Step 2.3.3.1.4.1

Multiply

$- 1$ by

$- 1$ .

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

Step 2.3.3.1.4.2

Multiply

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

$1$ .

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

Step 2.3.3.1.4.3

Multiply

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

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

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

Step 2.3.3.1.4.4

Raise

$\sin (x)$ to the power of

$1$ .

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

Step 2.3.3.1.4.5

Raise

$\sin (x)$ to the power of

$1$ .

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

Step 2.3.3.1.4.6

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3.3.1.4.7

Add

$1$ and

$1$ .

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

Step 2.3.3.2

Combine the numerators over the common denominator.

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

Step 2.3.4

Combine the numerators over the common denominator.

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

Step 2.3.5

Subtract

$\cos (x)$ from

$- \cos (x)$ .

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

Step 2.3.6

Factor using the perfect square rule.

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

Step 2.4

Simplify.

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Step 2.4.1

Rewrite

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

$(\cos (x) - 1) (\cos (x) - 1)$ .

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

Step 2.4.2

Expand

$(\cos (x) - 1) (\cos (x) - 1)$ using the FOIL Method.

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Step 2.4.2.1

Apply the distributive property.

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

Step 2.4.2.2

Apply the distributive property.

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

Step 2.4.2.3

Apply the distributive property.

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

Step 2.4.3

Simplify and combine like terms.

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

Step 3

Factor using the perfect square rule.

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

Step 4

Apply Pythagorean identity in reverse.

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

Step 5

Simplify.

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Step 5.1

Simplify the denominator.

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Step 5.1.1

Rewrite

$1$ as

$1^{2}$ .

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

Step 5.1.2

Since both terms are perfect squares, factor using the difference of squares formula,

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

$a = 1$ and

$b = \cos (x)$ .

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

Step 5.2

Cancel the common factor of

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

$1 - \cos (x)$ .

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

Step 6

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

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