Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Add $1$ and $1$ .

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

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

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

Step 2.2

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

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 2.3

Rewrite using the commutative property of multiplication.

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

Step 3

Multiply $- \sin (x) \sin (x)$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $1$ .

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

Step 4

Apply the cosine double-angle identity.

$\cos (2 x)$