Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

Combine.

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

Step 1.1.3

Apply the distributive property.

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

Step 1.1.4

Simplify terms.

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Step 1.1.4.1

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

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Step 1.1.4.1.1

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

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

Step 1.1.4.1.2

Cancel the common factor.

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

Step 1.1.4.1.3

Rewrite the expression.

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

Step 1.1.4.2

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

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

Step 1.1.4.3

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

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Step 1.1.4.3.1

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

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

Step 1.1.4.3.2

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

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

Step 1.1.4.4

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

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Step 1.1.4.4.1

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

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

Step 1.1.4.4.2

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

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

Step 1.1.4.4.3

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

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

Step 1.1.4.4.4

Reorder terms.

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

Step 1.1.4.4.5

Cancel the common factor.

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

Step 1.1.4.4.6

Rewrite the expression.

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

Step 1.1.4.5

Move the negative in front of the fraction.

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

Step 2

Simplify the right side.

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Step 2.1

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

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

Step 3

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

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

Step 4

Rewrite using the commutative property of multiplication.

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

Rewrite using the commutative property of multiplication.

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

$- 1 = - 1$

Step 8

Since $- 1 = - 1$ , the equation will always be true for any value of $x$ .

All real numbers

Step 9

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$