Trigonometry Examples

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

Step 1

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

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Step 1.1

Subtract $\csc (x) \cdot 1$ from both sides of the equation.

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

Multiply $- 1$ by $1$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Write each expression with a common denominator of $(1 - \cos (x)) \sin (x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

Reorder the factors of $(1 - \cos (x)) \sin (x)$ .

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify each term.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

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

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Step 2.6.1.1.1

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

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

Step 2.6.1.1.2

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

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

Step 2.6.1.1.3

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

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

Step 2.6.1.1.4

Add $1$ and $1$ .

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

Step 2.6.1.2

Apply the distributive property.

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

Step 2.6.1.3

Multiply $- 1$ by $1$ .

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

Step 2.6.1.4

Multiply $- - \cos (x)$ .

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Step 2.6.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.6.1.4.2

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

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

Step 2.6.1.5

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

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

Step 2.6.1.6

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

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

Step 2.6.1.7

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

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

Step 2.6.1.8

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

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

Step 2.6.1.9

Rewrite $- 1 (1 - \sin^{2} (x))$ as $- (1 - \sin^{2} (x))$ .

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

Step 2.6.1.10

Apply pythagorean identity.

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

Step 2.6.1.11

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

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Step 2.6.1.11.1

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

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

Step 2.6.1.11.2

Multiply by $1$ .

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

Step 2.6.1.11.3

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

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

Step 2.6.2

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

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Step 2.6.2.1

Reorder terms.

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

Step 2.6.2.2

Cancel the common factor.

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

Step 2.6.2.3

Rewrite the expression.

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

Step 2.7

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

$\cot (x) - \cos (x) = 0$

Step 3

Set the argument in $\cot (x)$ equal to $π n$ to find where the expression is undefined.

$x = π n$ , for any integer $n$

Step 4

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

${x | x = π n}$ $n$ , for any integer $n$

Step 5