Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

Rewrite $\frac{\sin (x)}{\cos (x)} - \sin (x) \cos (x)$ in a factored form.

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Step 1.2.1

Factor $\sin (x)$ out of $\frac{\sin (x)}{\cos (x)} - \sin (x) \cos (x)$ .

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Step 1.2.1.1

Factor $\sin (x)$ out of $\frac{\sin (x)}{\cos (x)}$ .

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

Step 1.2.1.2

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

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

Step 1.2.1.3

Factor $\sin (x)$ out of $\sin (x) \frac{1}{\cos (x)} + \sin (x) (- 1 \cos (x))$ .

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

Step 1.2.2

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

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

Step 2

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Apply the distributive property.

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

Step 6

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

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

Add $1$ and $1$ .

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

Step 8

Apply pythagorean identity.

$\sin^{2} (x)$