Simplify cos(x)sec(x)-(cos(x))/(sec(x))

$\cos (x) \sec (x) - \frac{\cos (x)}{\sec (x)}$

Step 1

Simplify each term.

Rewrite in terms of sines and cosines, then cancel the common factors.

Reorder $\cos (x)$ and $\sec (x)$ .

$\sec (x) \cos (x) - \frac{\cos (x)}{\sec (x)}$

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

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

Cancel the common factors.

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

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

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

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

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

Simplify.

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

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

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

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

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

$1 - {\cos (x)}^{1 + 1}$

Add $1$ and $1$ .

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

Step 2

Apply pythagorean identity.

$\sin^{2} (x)$