Trigonometry Examples

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

Step 1

Multiply both sides by $\cos (x) - 1$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

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

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Step 2.1.1.1

Cancel the common factor.

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

Step 2.1.1.2

Rewrite the expression.

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

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\frac{1 + \sec (x)}{1 - \sec (x)} (\cos (x) - 1)$ .

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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Step 2.2.1.3.1

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

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

Step 2.2.1.3.2

Combine.

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

Step 2.2.1.4

Apply the distributive property.

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

Step 2.2.1.5

Simplify by cancelling.

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Step 2.2.1.5.1

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

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Step 2.2.1.5.1.1

Cancel the common factor.

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

Step 2.2.1.5.1.2

Rewrite the expression.

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

Step 2.2.1.5.2

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

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Step 2.2.1.5.2.1

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

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

Step 2.2.1.5.2.2

Cancel the common factor.

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

Step 2.2.1.5.2.3

Rewrite the expression.

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

Step 2.2.1.6

Reduce the expression by cancelling the common factors.

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Step 2.2.1.6.1

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

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

Step 2.2.1.6.2

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

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

Step 2.2.1.6.3

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

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Step 2.2.1.6.3.1

Cancel the common factor.

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

Step 2.2.1.6.3.2

Rewrite the expression.

$\cos (x) + 1 = \cos (x) + 1$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms containing $\cos (x)$ to the left side of the equation.

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Step 3.1.1

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

$\cos (x) + 1 - \cos (x) = 1$

Step 3.1.2

Combine the opposite terms in $\cos (x) + 1 - \cos (x)$ .

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Step 3.1.2.1

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

$0 + 1 = 1$

Step 3.1.2.2

Add $0$ and $1$ .

$1 = 1$

Step 3.2

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

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$