Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Multiply $\frac{\cot (x)}{1 + \csc (x)}$ by $\frac{1 - \csc (x)}{1 - \csc (x)}$ .

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

Step 3

Combine.

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

Step 4

Simplify numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

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

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

Step 4.3

Reorder factors in $\cot (x) + \cot (x) (- \csc (x))$ .

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

Step 5

Simplify denominator.

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Step 5.1

Expand $(1 + \csc (x)) (1 - \csc (x))$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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

Step 6

Apply Pythagorean identity.

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Step 6.1

Reorder $1$ and $- \csc^{2} (x)$ .

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

Step 6.2

Factor $- 1$ out of $- \csc^{2} (x)$ .

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

Step 6.3

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 6.4

Factor $- 1$ out of $- \csc^{2} (x) - 1 \cdot - 1$ .

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

Step 6.5

Apply pythagorean identity.

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

Step 7

Convert to sines and cosines.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

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

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

Step 8

Simplify.

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Step 8.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2

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

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Step 8.2.1

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

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

Step 8.2.2

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

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

Step 8.2.3

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

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

Step 8.2.4

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

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

Step 8.2.5

Add $1$ and $1$ .

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

Step 8.3

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

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

Step 8.4

Write each expression with a common denominator of ${\sin (x)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.4.1

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

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

Step 8.4.2

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

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

Step 8.4.3

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

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

Step 8.4.4

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

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

Step 8.4.5

Add $1$ and $1$ .

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

Step 8.5

Combine the numerators over the common denominator.

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

Step 8.6

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

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Step 8.6.1

Factor $\cos (x)$ out of $\cos (x) \sin (x)$ .

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

Step 8.6.2

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

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

Step 8.6.3

Factor $\cos (x)$ out of $\cos (x) (\sin (x)) + \cos (x) \cdot - 1$ .

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

Step 8.7

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

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Step 8.7.1

Move the leading negative in $- \frac{{\sin (x)}^{2}}{{\cos (x)}^{2}}$ into the numerator.

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

Step 8.7.2

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

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

Step 8.7.3

Cancel the common factor.

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

Step 8.7.4

Rewrite the expression.

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

Step 8.8

Cancel the common factor of ${\sin (x)}^{2}$ .

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Step 8.8.1

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

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

Step 8.8.2

Cancel the common factor.

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

Step 8.8.3

Rewrite the expression.

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

Step 8.9

Move the negative in front of the fraction.

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

Step 8.10

Apply the distributive property.

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

Step 8.11

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

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

Step 8.12

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Reorder terms.

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

Step 11

Now consider the right side of the equation.

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

Step 12

Convert to sines and cosines.

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Step 12.1

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

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

Step 12.2

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

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

Step 13

Simplify.

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Step 13.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 13.2

Apply the distributive property.

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

Step 13.3

Cancel the common factor of $\sin (x)$ .

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Step 13.3.1

Cancel the common factor.

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

Step 13.3.2

Rewrite the expression.

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

Step 13.4

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

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

Step 14

Combine the numerators over the common denominator.

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

Step 15

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

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