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

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

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

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

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

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

Apply the distributive property.

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

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

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Cancel the common factor.

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

Rewrite the expression.

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

Rewrite using the commutative property of multiplication.

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

Multiply $- \frac{\cos (x)}{\sin (x)} \cos (x)$ .

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Combine $\cos (x)$ and $\frac{\cos (x)}{\sin (x)}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Simplify each term.

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Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

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

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

$\csc (x) - \frac{\cos (x) \cos (x)}{\sin (x)}$

Separate fractions.

$\csc (x) - (\frac{\cos (x)}{1} \cdot \frac{\cos (x)}{\sin (x)})$

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

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

$\csc (x) - \cos (x) \cot (x)$

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

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