Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

Simplify the left side.

Tap for more steps...

Step 2.1.1

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

Tap for more steps...

Step 2.1.1.1

Cancel the common factor.

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

Step 2.1.1.2

Rewrite the expression.

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

Step 2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1

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

Tap for more steps...

Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

Tap for more steps...

Step 2.2.1.3.1

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

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

Step 2.2.1.3.2

Combine.

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

Step 2.2.1.4

Apply the distributive property.

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

Step 2.2.1.5

Simplify by cancelling.

Tap for more steps...

Step 2.2.1.5.1

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

Tap for more steps...

Step 2.2.1.5.1.1

Cancel the common factor.

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

Step 2.2.1.5.1.2

Rewrite the expression.

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

Step 2.2.1.5.2

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

Tap for more steps...

Step 2.2.1.5.2.1

Cancel the common factor.

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

Step 2.2.1.5.2.2

Rewrite the expression.

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

Step 2.2.1.6

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

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

Step 2.2.1.7

Simplify the denominator.

Tap for more steps...

Step 2.2.1.7.1

Move $- 1$ to the left of $\cos (x)$ .

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

Step 2.2.1.7.2

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

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

Step 2.2.1.8

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

Tap for more steps...

Step 2.2.1.8.1

Cancel the common factor.

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

Step 2.2.1.8.2

Rewrite the expression.

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

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

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

Tap for more steps...

Step 3.1.2.1

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

$1 + 0 = 1$

Step 3.1.2.2

Add $1$ and $0$ .

$1 = 1$

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:

$(- \infty, \infty)$