Simplify (cos(2x))/(cos(x)^2)

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

Step 1

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

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

Step 2

Separate fractions.

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

Step 3

Rewrite $\frac{\cos (2 x)}{\cos (x)}$ as a product.

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

Step 4

Write $\cos (2 x)$ as a fraction with denominator $1$ .

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

Step 5

Simplify.

Divide $\cos (2 x)$ by $1$ .

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

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 6

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x) (\cos (2 x) \sec (x))$

Step 7

Multiply $\sec (x) (\cos (2 x) \sec (x))$ .

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

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

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

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

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

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

Add $1$ and $1$ .

$\sec^{2} (x) \cos (2 x)$