Trigonometry Examples

$\sec (2 x) = \frac{\sec^{2} (x)}{2 - \sec^{2} (x)}$

Step 1

Start on the right side.

$\frac{\sec^{2} (x)}{2 - \sec^{2} (x)}$

Step 2

Convert to sines and cosines.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Simplify.

Tap for more steps...

Step 2.3.1

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 2.3.2

One to any power is one.

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

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 2.4.2

One to any power is one.

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

Simplify the denominator.

Tap for more steps...

Step 3.2.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{{\cos (x)}^{2}}{{\cos (x)}^{2}}$ .

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

Step 3.2.2

Combine the numerators over the common denominator.

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

Step 3.3

Combine.

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

Step 3.4

Multiply $1$ by $1$ .

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

Step 3.5

Combine ${\cos (x)}^{2}$ and $\frac{2 {\cos (x)}^{2} - 1}{{\cos (x)}^{2}}$ .

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

Step 3.6

Reduce the expression by cancelling the common factors.

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

Step 4

Apply the cosine double-angle identity.

$\frac{1}{\cos (2 x)}$

Step 5

Rewrite $\frac{1}{\cos (2 x)}$ as $\sec (2 x)$ .

$\sec (2 x)$

Step 6

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

$\sec (2 x) = \frac{\sec^{2} (x)}{2 - \sec^{2} (x)}$ is an identity