Evaluate (cos(x))/(1+sin(x))+(1+sin(x))/(cos(x))

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

To write $\frac{\cos (x)}{1 + \sin (x)}$ as a fraction with a common denominator, multiply by $\frac{\cos (x)}{\cos (x)}$ .

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

To write $\frac{1 + \sin (x)}{\cos (x)}$ as a fraction with a common denominator, multiply by $\frac{1 + \sin (x)}{1 + \sin (x)}$ .

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

Write each expression with a common denominator of $(1 + \sin (x)) \cos (x)$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

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

Reorder the factors of $(1 + \sin (x)) \cos (x)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Expand $(1 + \sin (x)) (1 + \sin (x))$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

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

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

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

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

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

Multiply $\sin (x) \sin (x)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

Add $\sin (x)$ and $\sin (x)$ .

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

Rewrite $\cos^{2} (x) + 1 + 2 \sin (x) + \sin^{2} (x)$ in a factored form.

Regroup terms.

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

Rearrange terms.

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

Apply pythagorean identity.

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

Add $1$ and $1$ .

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

Factor $2$ out of $2 + 2 \sin (x)$ .

Factor $2$ out of $2$ .

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

Factor $2$ out of $2 \cdot 1 + 2 \sin (x)$ .

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

Cancel the common factor of $1 + \sin (x)$ .

Cancel the common factor.

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

Rewrite the expression.

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

Separate fractions.

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

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

$\frac{2}{1} \sec (x)$

Divide $2$ by $1$ .

$2 \sec (x)$

Evaluate (cos(x))/(1+sin(x))+(1+sin(x))/(cos(x))

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