Trigonometry Examples

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

Step 1

Start on the right side.

$\frac{1}{\csc (x) + \cot (x)}$

Step 2

Convert to sines and cosines.

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Step 2.1

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

$\frac{1}{\frac{1}{\sin (x)} + \cot (x)}$

Step 2.2

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

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

Step 3

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

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

Step 4

Combine.

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

Step 5

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

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

Step 6

Simplify denominator.

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Step 6.1

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 6.1.1

Apply the distributive property.

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

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Apply the distributive property.

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

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

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Step 6.2.1.1.1

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

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

Step 6.2.1.1.2

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

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

Step 6.2.1.1.3

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

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

Step 6.2.1.1.4

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

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

Step 6.2.1.1.5

Add $1$ and $1$ .

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

Step 6.2.1.2

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

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Step 6.2.1.2.1

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

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

Step 6.2.1.2.2

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

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

Step 6.2.1.2.3

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

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

Step 6.2.1.2.4

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

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

Step 6.2.1.2.5

Add $1$ and $1$ .

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

Step 6.2.1.3

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

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Step 6.2.1.3.1

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

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

Step 6.2.1.3.2

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

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

Step 6.2.1.3.3

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

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

Step 6.2.1.3.4

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

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

Step 6.2.1.3.5

Add $1$ and $1$ .

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

Step 6.2.1.4

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

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Step 6.2.1.4.1

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

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

Step 6.2.1.4.2

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

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

Step 6.2.1.4.3

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

Step 6.2.1.4.4

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

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

Step 6.2.1.4.5

Add $1$ and $1$ .

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

Step 6.2.1.4.6

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

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

Step 6.2.1.4.7

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

Step 6.2.1.4.8

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

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

Step 6.2.1.4.9

Add $1$ and $1$ .

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

Step 6.2.2

Combine the numerators over the common denominator.

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

Step 6.3

Combine the numerators over the common denominator.

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

Step 6.4

Add $\cos (x)$ and $\cos (x)$ .

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

Step 6.5

Factor using the perfect square rule.

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

Step 7

Apply Pythagorean identity in reverse.

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

Step 8

Simplify.

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Step 8.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2

Simplify the numerator.

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Step 8.2.1

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

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

Step 8.2.2

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 $b = \cos (x)$ .

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

Step 8.3

Cancel the common factors.

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Step 8.3.1

Factor $1 + \cos (x)$ out of ${(1 + \cos (x))}^{2}$ .

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

Step 8.3.2

Cancel the common factor.

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

Step 8.3.3

Rewrite the expression.

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

Step 8.4

Apply the distributive property.

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

Step 8.5

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

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

Step 8.6

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

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

Step 8.7

Combine the numerators over the common denominator.

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

Step 8.8

Simplify each term.

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Step 8.8.1

Apply the distributive property.

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

Step 8.8.2

Multiply $\cos (x)$ by $1$ .

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

Step 8.8.3

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

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Step 8.8.3.1

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

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

Step 8.8.3.2

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

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

Step 8.8.3.3

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

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

Step 8.8.3.4

Add $1$ and $1$ .

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

Step 8.9

Add $- \cos (x)$ and $\cos (x)$ .

$\frac{1 + 0 - {\cos (x)}^{2}}{(1 + \cos (x)) \sin (x)}$

Step 8.10

Add $1$ and $0$ .

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

Step 8.11

Simplify the numerator.

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Step 8.11.1

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

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

Step 8.11.2

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 $b = \cos (x)$ .

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

Step 8.12

Cancel the common factor of $1 + \cos (x)$ .

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

Step 9

Now consider the left side of the equation.

$\csc (x) - \cot (x)$

Step 10

Convert to sines and cosines.

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Step 10.1

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

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

Step 10.2

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

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

Step 11

Combine the numerators over the common denominator.

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

Step 12

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

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