Precalculus Examples

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

Step 1

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

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

Step 2

Simplify each term.

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Step 2.1

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

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

Step 2.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 2.3

One to any power is one.

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

Step 3

Expand $(\cos (x) + \frac{1}{\cos (x)}) (\sin^{2} (x) + \frac{1}{\cos^{2} (x)})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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Step 4.1.1.1

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

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

Step 4.1.1.2

Cancel the common factor.

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

Step 4.1.1.3

Rewrite the expression.

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

Step 4.1.2

Combine $\frac{1}{\cos (x)}$ and $\sin^{2} (x)$ .

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

Step 4.1.3

Combine.

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

Step 4.1.4

Multiply $\cos (x)$ by $\cos^{2} (x)$ by adding the exponents.

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Step 4.1.4.1

Multiply $\cos (x)$ by $\cos^{2} (x)$ .

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Step 4.1.4.1.1

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

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

Step 4.1.4.1.2

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

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

Step 4.1.4.2

Add $1$ and $2$ .

$\cos (x) \sin^{2} (x) + \frac{1}{\cos (x)} + \frac{\sin^{2} (x)}{\cos (x)} + \frac{1 \cdot 1}{\cos^{3} (x)}$

Step 4.1.5

Multiply $1$ by $1$ .

$\cos (x) \sin^{2} (x) + \frac{1}{\cos (x)} + \frac{\sin^{2} (x)}{\cos (x)} + \frac{1}{\cos^{3} (x)}$

Step 4.2

Simplify each term.

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Step 4.2.1

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\cos (x) \sin^{2} (x) + \sec (x) + \frac{\sin^{2} (x)}{\cos (x)} + \frac{1}{\cos^{3} (x)}$

Step 4.2.2

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

$\cos (x) \sin^{2} (x) + \sec (x) + \frac{\sin (x) \sin (x)}{\cos (x)} + \frac{1}{\cos^{3} (x)}$

Step 4.2.3

Separate fractions.

$\cos (x) \sin^{2} (x) + \sec (x) + \frac{\sin (x)}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos^{3} (x)}$

Step 4.2.4

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

$\cos (x) \sin^{2} (x) + \sec (x) + \frac{\sin (x)}{1} \tan (x) + \frac{1}{\cos^{3} (x)}$

Step 4.2.5

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

$\cos (x) \sin^{2} (x) + \sec (x) + \sin (x) \tan (x) + \frac{1}{\cos^{3} (x)}$