Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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Step 1.1.1

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 1.1.2

Rewrite $\csc (x)$ in terms of sines and cosines.

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

Step 1.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.4

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

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

Step 2

Simplify the right side.

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Step 2.1

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

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Step 2.1.1

Rewrite $\csc (x)$ in terms of sines and cosines.

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

Step 2.1.2

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 2.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.4

Apply the distributive property.

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

Step 2.1.5

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

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Step 2.1.5.1

Cancel the common factor.

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

Step 2.1.5.2

Rewrite the expression.

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

Step 2.1.6

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

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

Step 3

Multiply both sides of the equation by $\cos (x)$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 5

Apply the distributive property.

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

Step 6

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

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

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

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

Move all the expressions to the left side of the equation.

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Step 8.1

Subtract $1$ from both sides of the equation.

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

Step 8.2

Subtract $\sin (x)$ from both sides of the equation.

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

Step 9

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

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Step 9.1

Simplify each term.

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Step 9.1.1

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

$\frac{\cos (x) \cos (x)}{\sin (x) (\frac{1}{\sin (x)} - 1)} - 1 - \sin (x) = 0$

Step 9.1.2

Separate fractions.

$\frac{\cos (x)}{\frac{1}{\sin (x)} - 1} \cdot \frac{\cos (x)}{\sin (x)} - 1 - \sin (x) = 0$

Step 9.1.3

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\frac{\cos (x)}{\frac{1}{\sin (x)} - 1} \cot (x) - 1 - \sin (x) = 0$

Step 9.1.4

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\frac{\cos (x)}{\csc (x) - 1} \cot (x) - 1 - \sin (x) = 0$

Step 9.1.5

Combine $\frac{\cos (x)}{\csc (x) - 1}$ and $\cot (x)$ .

$\frac{\cos (x) \cot (x)}{\csc (x) - 1} - 1 - \sin (x) = 0$

Step 9.2

To write $- \sin (x)$ as a fraction with a common denominator, multiply by $\frac{\csc (x) - 1}{\csc (x) - 1}$ .

$\frac{\cos (x) \cot (x)}{\csc (x) - 1} - \sin (x) \cdot \frac{\csc (x) - 1}{\csc (x) - 1} - 1 = 0$

Step 9.3

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

$\frac{\cos (x) \cot (x)}{\csc (x) - 1} + \frac{- \sin (x) (\csc (x) - 1)}{\csc (x) - 1} - 1 = 0$

Step 9.4

Combine the numerators over the common denominator.

$\frac{\cos (x) \cot (x) - \sin (x) (\csc (x) - 1)}{\csc (x) - 1} - 1 = 0$

Step 9.5

Simplify the numerator.

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Step 9.5.1

Apply the distributive property.

$\frac{\cos (x) \cot (x) - \sin (x) \csc (x) - \sin (x) \cdot - 1}{\csc (x) - 1} - 1 = 0$

Step 9.5.2

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 9.5.2.1

Add parentheses.

$\frac{\cos (x) \cot (x) - (\sin (x) \csc (x)) - \sin (x) \cdot - 1}{\csc (x) - 1} - 1 = 0$

Step 9.5.2.2

Reorder $\sin (x)$ and $\csc (x)$ .

$\frac{\cos (x) \cot (x) - (\csc (x) \sin (x)) - \sin (x) \cdot - 1}{\csc (x) - 1} - 1 = 0$

Step 9.5.2.3

Rewrite $- \sin (x) \csc (x)$ in terms of sines and cosines.

$\frac{\cos (x) \cot (x) - (\frac{1}{\sin (x)} \sin (x)) - \sin (x) \cdot - 1}{\csc (x) - 1} - 1 = 0$

Step 9.5.2.4

Cancel the common factors.

$\frac{\cos (x) \cot (x) - 1 \cdot 1 - \sin (x) \cdot - 1}{\csc (x) - 1} - 1 = 0$

Step 9.5.3

Multiply $- \sin (x) \cdot - 1$ .

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Step 9.5.3.1

Multiply $- 1$ by $- 1$ .

$\frac{\cos (x) \cot (x) - 1 \cdot 1 + 1 \sin (x)}{\csc (x) - 1} - 1 = 0$

Step 9.5.3.2

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

$\frac{\cos (x) \cot (x) - 1 \cdot 1 + \sin (x)}{\csc (x) - 1} - 1 = 0$

Step 9.5.4

Multiply $- 1$ by $1$ .

$\frac{\cos (x) \cot (x) - 1 + \sin (x)}{\csc (x) - 1} - 1 = 0$

Step 10

Divide each term in the equation by $\cos (x)$ .

$\frac{\frac{\cos (x) \cot (x) - 1 + \sin (x)}{\csc (x) - 1}}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 11

Multiply the numerator by the reciprocal of the denominator.

$\frac{\cos (x) \cot (x) - 1 + \sin (x)}{\csc (x) - 1} \cdot \frac{1}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 12

Simplify the numerator.

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Step 12.1

Rewrite $\cot (x)$ in terms of sines and cosines.

$\frac{\cos (x) (\frac{\cos (x)}{\sin (x)}) - 1 + \sin (x)}{\csc (x) - 1} \cdot \frac{1}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 12.2

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

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Step 12.2.1

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

$\frac{\frac{\cos (x) \cos (x)}{\sin (x)} - 1 + \sin (x)}{\csc (x) - 1} \cdot \frac{1}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 12.2.2

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

$\frac{\frac{\cos (x) \cos (x)}{\sin (x)} - 1 + \sin (x)}{\csc (x) - 1} \cdot \frac{1}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 12.2.3

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

$\frac{\frac{\cos (x) \cos (x)}{\sin (x)} - 1 + \sin (x)}{\csc (x) - 1} \cdot \frac{1}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 12.2.4

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

$\frac{\frac{{\cos (x)}^{1 + 1}}{\sin (x)} - 1 + \sin (x)}{\csc (x) - 1} \cdot \frac{1}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 12.2.5

Add $1$ and $1$ .

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

Step 13

Rewrite $\csc (x)$ in terms of sines and cosines.

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

Step 14

Multiply the numerator and denominator of the fraction by $\sin (x)$ .

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Step 14.1

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

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

Step 14.2

Combine.

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

Step 15

Apply the distributive property.

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

Step 16

Simplify by cancelling.

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Step 16.1

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

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Step 16.1.1

Cancel the common factor.

Step 16.1.2

Rewrite the expression.

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

Step 16.2

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

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Step 16.2.1

Cancel the common factor.

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

Step 16.2.2

Rewrite the expression.

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

Step 17

Simplify the numerator.

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Step 17.1

Rearrange terms.

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

Step 17.2

Rearrange terms.

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

Step 17.3

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

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

Step 17.4

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

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

Step 17.5

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

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

Step 17.6

Add $1$ and $1$ .

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

Step 17.7

Apply pythagorean identity.

$\frac{\sin (x) \cdot - 1 + 1}{1 + \sin (x) \cdot - 1} \cdot \frac{1}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 17.8

Move $- 1$ to the left of $\sin (x)$ .

$\frac{- 1 \cdot \sin (x) + 1}{1 + \sin (x) \cdot - 1} \cdot \frac{1}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 17.9

Rewrite $- 1 \sin (x)$ as $- \sin (x)$ .

$\frac{- \sin (x) + 1}{1 + \sin (x) \cdot - 1} \cdot \frac{1}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 18

Simplify the denominator.

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Step 18.1

Move $- 1$ to the left of $\sin (x)$ .

$\frac{- \sin (x) + 1}{1 - 1 \cdot \sin (x)} \cdot \frac{1}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 18.2

Rewrite $- 1 \sin (x)$ as $- \sin (x)$ .

$\frac{- \sin (x) + 1}{1 - \sin (x)} \cdot \frac{1}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 19

Reduce the expression by cancelling the common factors.

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Step 19.1

Cancel the common factor of $- \sin (x) + 1$ and $1 - \sin (x)$ .

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Step 19.1.1

Reorder terms.

$\frac{1 - \sin (x)}{1 - \sin (x)} \cdot \frac{1}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 19.1.2

Cancel the common factor.

$\frac{1 - \sin (x)}{1 - \sin (x)} \cdot \frac{1}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 19.1.3

Rewrite the expression.

$1 (\frac{1}{\cos (x)}) + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 19.2

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

$\frac{1}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 20

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x) + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 21

Separate fractions.

$\sec (x) + \frac{- 1}{1} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 22

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x) + \frac{- 1}{1} \cdot \sec (x) = \frac{0}{\cos (x)}$

Step 23

Divide $- 1$ by $1$ .

$\sec (x) - \sec (x) = \frac{0}{\cos (x)}$

Step 24

Separate fractions.

$\sec (x) - \sec (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 25

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x) - \sec (x) = \frac{0}{1} \cdot \sec (x)$

Step 26

Divide $0$ by $1$ .

$\sec (x) - \sec (x) = 0 \sec (x)$

Step 27

Multiply $0$ by $\sec (x)$ .

$\sec (x) - \sec (x) = 0$

Step 28

Subtract $\sec (x)$ from $\sec (x)$ .

$0 = 0$

Step 29

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 30

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$