Precalculus Examples

$\sin (x) \tan (x) - \csc (x) \tan (x)$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

Add $1$ and $1$ .

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

Step 1.1.3

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

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Step 1.1.3.1

Add parentheses.

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

Step 1.1.3.2

Reorder $\csc (x)$ and $\tan (x)$ .

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

Step 1.1.3.3

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

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

Step 1.1.3.4

Cancel the common factors.

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

Step 1.2

Simplify terms.

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Step 1.2.1

Combine the numerators over the common denominator.

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 2

Apply pythagorean identity.

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

Step 3

Cancel the common factor of $\cos^{2} (x)$ and $\cos (x)$ .

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Step 3.1

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

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Multiply by $1$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.2.4

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

$- \cos (x)$