Trigonometry Examples

$\frac{\cos (x) (\tan (x) - \sec (x))}{1 - \csc (x)} = \sin (x)$

Step 1

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

$\frac{\frac{\cos (x) (\tan (x) - \sec (x))}{1 - \csc (x)}}{\cos (x)} = \frac{\sin (x)}{\cos (x)}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{\cos (x) (\tan (x) - \sec (x))}{1 - \csc (x)} \cdot \frac{1}{\cos (x)} = \frac{\sin (x)}{\cos (x)}$

Step 3

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

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Step 3.1

Cancel the common factor.

$\frac{\cos (x) (\tan (x) - \sec (x))}{1 - \csc (x)} \cdot \frac{1}{\cos (x)} = \frac{\sin (x)}{\cos (x)}$

Step 3.2

Rewrite the expression.

$\frac{\tan (x) - \sec (x)}{1 - \csc (x)} = \frac{\sin (x)}{\cos (x)}$

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Combine.

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

Step 7

Apply the distributive property.

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

Step 8

Simplify by cancelling.

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Step 8.1

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

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Step 8.1.1

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

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

Step 8.1.2

Cancel the common factor.

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

Step 8.1.3

Rewrite the expression.

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Add $1$ and $1$ .

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

Step 8.6

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

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Step 8.6.1

Move the leading negative in $- \frac{1}{\cos (x)}$ into the numerator.

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

Step 8.6.2

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

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

Step 8.6.3

Cancel the common factor.

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

Step 8.6.4

Rewrite the expression.

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

Step 8.7

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

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Step 8.7.1

Move the leading negative in $- \frac{1}{\sin (x)}$ into the numerator.

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

Step 8.7.2

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

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

Step 8.7.3

Cancel the common factor.

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

Step 8.7.4

Rewrite the expression.

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 10

Simplify the denominator.

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Step 10.1

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

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Step 10.1.1

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

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

Step 10.1.2

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

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

Step 10.1.3

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

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

Step 10.2

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

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

Step 11

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

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Step 11.1

Cancel the common factor.

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

Step 11.2

Rewrite the expression.

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

Step 12

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

$\tan (x) = \frac{\sin (x)}{\cos (x)}$

Step 13

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

$\tan (x) = \tan (x)$

Step 14

For the two functions to be equal, the arguments of each must be equal.

$x = x$

Step 15

Move all terms containing $x$ to the left side of the equation.

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Step 15.1

Subtract $x$ from both sides of the equation.

$x - x = 0$

Step 15.2

Subtract $x$ from $x$ .

$0 = 0$

Step 16

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

All real numbers

Step 17

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$