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

$\cos (x) (\cos (x) - \sec (x))$

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

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

Apply the distributive property.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Simplify terms.

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Rewrite using the commutative property of multiplication.

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

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

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Factor $\cos (x)$ out of $- \cos (x)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Rewrite $- 1$ as $- 1 (1)$ .

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

Factor $- 1$ out of $\cos^{2} (x)$ .

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

Factor $- 1$ out of $- 1 (1) - 1 (- \cos^{2} (x))$ .

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

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

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

Apply pythagorean identity.

$- \sin^{2} (x)$

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

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