Calculus Examples

${(1 + \frac{\cos (x)}{4})}^{2}$

Step 1

Rewrite ${(1 + \frac{\cos (x)}{4})}^{2}$ as $(1 + \frac{\cos (x)}{4}) (1 + \frac{\cos (x)}{4})$ .

$(1 + \frac{\cos (x)}{4}) (1 + \frac{\cos (x)}{4})$

Step 2

Expand $(1 + \frac{\cos (x)}{4}) (1 + \frac{\cos (x)}{4})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$1 (1 + \frac{\cos (x)}{4}) + \frac{\cos (x)}{4} (1 + \frac{\cos (x)}{4})$

Step 2.2

Apply the distributive property.

$1 \cdot 1 + 1 \frac{\cos (x)}{4} + \frac{\cos (x)}{4} (1 + \frac{\cos (x)}{4})$

Step 2.3

Apply the distributive property.

$1 \cdot 1 + 1 \frac{\cos (x)}{4} + \frac{\cos (x)}{4} \cdot 1 + \frac{\cos (x)}{4} \cdot \frac{\cos (x)}{4}$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $1$ by $1$ .

$1 + 1 \frac{\cos (x)}{4} + \frac{\cos (x)}{4} \cdot 1 + \frac{\cos (x)}{4} \cdot \frac{\cos (x)}{4}$

Step 3.1.2

Multiply $\frac{\cos (x)}{4}$ by $1$ .

$1 + \frac{\cos (x)}{4} + \frac{\cos (x)}{4} \cdot 1 + \frac{\cos (x)}{4} \cdot \frac{\cos (x)}{4}$

Step 3.1.3

Multiply $\frac{\cos (x)}{4}$ by $1$ .

$1 + \frac{\cos (x)}{4} + \frac{\cos (x)}{4} + \frac{\cos (x)}{4} \cdot \frac{\cos (x)}{4}$

Step 3.1.4

Multiply $\frac{\cos (x)}{4} \cdot \frac{\cos (x)}{4}$ .

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Step 3.1.4.1

Multiply $\frac{\cos (x)}{4}$ by $\frac{\cos (x)}{4}$ .

$1 + \frac{\cos (x)}{4} + \frac{\cos (x)}{4} + \frac{\cos (x) \cos (x)}{4 \cdot 4}$

Step 3.1.4.2

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

$1 + \frac{\cos (x)}{4} + \frac{\cos (x)}{4} + \frac{\cos^{1} (x) \cos (x)}{4 \cdot 4}$

Step 3.1.4.3

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

$1 + \frac{\cos (x)}{4} + \frac{\cos (x)}{4} + \frac{\cos^{1} (x) \cos^{1} (x)}{4 \cdot 4}$

Step 3.1.4.4

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

$1 + \frac{\cos (x)}{4} + \frac{\cos (x)}{4} + \frac{{\cos (x)}^{1 + 1}}{4 \cdot 4}$

Step 3.1.4.5

Add $1$ and $1$ .

$1 + \frac{\cos (x)}{4} + \frac{\cos (x)}{4} + \frac{\cos^{2} (x)}{4 \cdot 4}$

Step 3.1.4.6

Multiply $4$ by $4$ .

$1 + \frac{\cos (x)}{4} + \frac{\cos (x)}{4} + \frac{\cos^{2} (x)}{16}$

Step 3.2

Add $\frac{\cos (x)}{4}$ and $\frac{\cos (x)}{4}$ .

$1 + 2 \frac{\cos (x)}{4} + \frac{\cos^{2} (x)}{16}$

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Factor $2$ out of $4$ .

$1 + 2 \frac{\cos (x)}{2 (2)} + \frac{\cos^{2} (x)}{16}$

Step 4.2

Cancel the common factor.

$1 + 2 \frac{\cos (x)}{2 \cdot 2} + \frac{\cos^{2} (x)}{16}$

Step 4.3

Rewrite the expression.

$1 + \frac{\cos (x)}{2} + \frac{\cos^{2} (x)}{16}$