Precalculus Examples

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

Step 1

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

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

Step 2

Rewrite ${(\frac{\sin (x)}{\cos (x)} + 1)}^{2}$ as $(\frac{\sin (x)}{\cos (x)} + 1) (\frac{\sin (x)}{\cos (x)} + 1)$ .

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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Step 4.1.1.1

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

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

Step 4.1.1.2

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

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

Step 4.1.1.3

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

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

Step 4.1.1.4

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

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

Step 4.1.1.5

Add $1$ and $1$ .

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

Step 4.1.1.6

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

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

Step 4.1.1.7

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

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

Step 4.1.1.8

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

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

Step 4.1.1.9

Add $1$ and $1$ .

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

Step 4.1.2

Multiply $\frac{\sin (x)}{\cos (x)}$ by $1$ .

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

Step 4.1.3

Multiply $\frac{\sin (x)}{\cos (x)}$ by $1$ .

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

Step 4.1.4

Multiply $1$ by $1$ .

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

Step 4.2

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

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

Step 5

Combine $2$ and $\frac{\sin (x)}{\cos (x)}$ .

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

Step 6

Apply the distributive property.

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

Step 7

Simplify.

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Step 7.1

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

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Step 7.1.1

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

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

Step 7.1.2

Cancel the common factor.

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

Step 7.1.3

Rewrite the expression.

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

Step 7.2

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

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

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

Step 7.3

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

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

Step 8

Simplify each term.

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Step 8.1

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

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

Step 8.2

Separate fractions.

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

Step 8.3

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

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

Step 8.4

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

$\sin (x) \tan (x) + 2 \sin (x) + \cos (x)$