Trigonometry Examples

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

Step 1

Start on the left side.

$\frac{\sec (x)}{1 + \sec (x)}$

Step 2

Convert to sines and cosines.

Tap for more steps...

Step 2.1

Apply the reciprocal identity to $\sec (x)$ .

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

Step 2.2

Apply the reciprocal identity to $\sec (x)$ .

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

Simplify the denominator.

Tap for more steps...

Step 3.2.1

Write $1$ as a fraction with a common denominator.

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

Step 3.2.2

Combine the numerators over the common denominator.

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

Step 3.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.4

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

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

Step 3.5

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

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

Step 4

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

$\frac{\sec (x)}{1 + \sec (x)} = \frac{1}{\cos (x) + 1}$ is an identity