Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify each term.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Simplify the numerator.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Add $1$ and $1$ .

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

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5

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

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

Step 3

Apply Pythagorean identity in reverse.

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

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Simplify the numerator.

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Step 4.1.1.1

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

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

Step 4.1.1.2

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 $b = \sin (x)$ .

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

Step 4.1.2

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

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Step 4.1.2.1

Cancel the common factor.

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

Step 4.1.2.2

Rewrite the expression.

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 4.5

Simplify the numerator.

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

Step 5

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

$\csc (x)$

Step 6

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

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