Simplify (sin(theta))/(1+cos(theta))+(1-cos(theta))/(sin(theta))

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

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

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Multiply $\sin (θ) \sin (θ)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

Expand $(1 - \cos (θ)) (1 + \cos (θ))$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

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

Multiply $\cos (θ)$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

Multiply $- \cos (θ) \cos (θ)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

Subtract $\cos (θ)$ from $\cos (θ)$ .

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

Add $1$ and $0$ .

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

Apply pythagorean identity.

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

Add $\sin^{2} (θ)$ and $\sin^{2} (θ)$ .

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

Step 6

Cancel the common factor of $\sin^{2} (θ)$ and $\sin (θ)$ .

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

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 \sin (θ)}{1 + \cos (θ)}$

Simplify (sin(theta))/(1+cos(theta))+(1-cos(theta))/(sin(theta))

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