Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Simplify.

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Step 2.3.1

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

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

Step 2.3.2

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

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Step 2.3.2.1

Apply the distributive property.

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

Step 2.3.2.2

Apply the distributive property.

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

Step 2.3.2.3

Apply the distributive property.

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

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

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Step 2.3.3.1.1.1

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

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

Step 2.3.3.1.1.2

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

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

Step 2.3.3.1.1.3

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

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

Step 2.3.3.1.1.4

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

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

Step 2.3.3.1.1.5

Add $1$ and $1$ .

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

Step 2.3.3.1.2

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

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Step 2.3.3.1.2.1

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

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

Step 2.3.3.1.2.2

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

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

Step 2.3.3.1.2.3

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

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

Step 2.3.3.1.2.4

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

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

Step 2.3.3.1.2.5

Add $1$ and $1$ .

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

Step 2.3.3.1.3

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

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Step 2.3.3.1.3.1

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

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

Step 2.3.3.1.3.2

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

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

Step 2.3.3.1.3.3

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

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

Step 2.3.3.1.3.4

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

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

Step 2.3.3.1.3.5

Add $1$ and $1$ .

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

Step 2.3.3.1.4

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

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Step 2.3.3.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.3.1.4.2

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

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

Step 2.3.3.1.4.3

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

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

Step 2.3.3.1.4.4

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

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

Step 2.3.3.1.4.5

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

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

Step 2.3.3.1.4.6

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

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

Step 2.3.3.1.4.7

Add $1$ and $1$ .

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

Step 2.3.3.1.4.8

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

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

Step 2.3.3.1.4.9

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

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

Step 2.3.3.1.4.10

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

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

Step 2.3.3.1.4.11

Add $1$ and $1$ .

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

Step 2.3.3.2

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

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

Step 2.3.4

Simplify each term.

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Step 2.3.4.1

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

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

Step 2.3.4.2

Move the negative in front of the fraction.

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

Step 2.3.5

Combine the numerators over the common denominator.

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

Step 3

Simplify the expression.

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Step 3.1

Factor $\cos (x)$ out of $- 2 \cos (x) + \cos^{2} (x)$ .

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Step 3.1.1

Factor $\cos (x)$ out of $- 2 \cos (x)$ .

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

Step 3.1.2

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

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

Step 3.1.3

Factor $\cos (x)$ out of $\cos (x) \cdot - 2 + \cos (x) \cos (x)$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 3.3

Simplify the numerator.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Move $- 2$ to the left of $\cos (x)$ .

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

Step 3.3.3

Multiply $\cos (x) \cos (x)$ .

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Step 3.3.3.1

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

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

Step 3.3.3.2

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

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

Step 3.3.3.3

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

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

Step 3.3.3.4

Add $1$ and $1$ .

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

Step 3.3.4

Factor using the perfect square rule.

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Step 3.3.4.1

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

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

Step 3.3.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 \cos (x) = 2 \cdot 1 \cdot \cos (x)$

Step 3.3.4.3

Rewrite the polynomial.

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

Step 3.3.4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 1$ and $b = \cos (x)$ .

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

Step 4

Apply Pythagorean identity in reverse.

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

Step 5

Simplify.

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Step 5.1

Simplify the denominator.

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Step 5.1.1

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

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

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 $b = \cos (x)$ .

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

Step 5.2

Cancel the common factor of ${(1 - \cos (x))}^{2}$ 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.

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