Precalculus Examples

$(\sin (x) - 1) (\tan (x) + \sec (x)) = - \cos (x)$

Step 1

Start on the left side.

$(\sin (x) - 1) (\tan (x) + \sec (x))$

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

Expand $(\sin (x) - 1) (\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

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

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Step 2.3.1.1.1

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

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

Step 2.3.1.1.2

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

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

Step 2.3.1.1.3

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

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

Step 2.3.1.1.4

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

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

Step 2.3.1.1.5

Add $1$ and $1$ .

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

Step 2.3.1.2

Combine $\sin (x)$ and $\frac{1}{\cos (x)}$ .

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

Step 2.3.1.3

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

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

Step 2.3.1.4

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

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

Step 2.3.2

Subtract $\frac{\sin (x)}{\cos (x)}$ from $\frac{\sin (x)}{\cos (x)}$ .

$\frac{\sin^{2} (x)}{\cos (x)} + 0 - \frac{1}{\cos (x)}$

Step 2.3.3

Add $\frac{\sin^{2} (x)}{\cos (x)}$ and $0$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

Apply pythagorean identity.

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

Step 2.11

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

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Step 2.11.1

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

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

Step 2.11.2

Cancel the common factors.

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Step 2.11.2.1

Multiply by $1$ .

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

Step 2.11.2.2

Cancel the common factor.

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

Step 2.11.2.3

Rewrite the expression.

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

Step 2.11.2.4

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

$- \cos (x)$

Step 3

Because the two sides have been shown to be equivalent, the equation is an identity.

$(\sin (x) - 1) (\tan (x) + \sec (x)) = - \cos (x)$ is an identity