Simplify (csc(theta)+cot(theta))(1-cos(theta))

$(\csc (θ) + \cot (θ)) (1 - \cos (θ))$

Simplify each term.

Rewrite $\csc (θ)$ in terms of sines and cosines.

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

Rewrite $\cot (θ)$ in terms of sines and cosines.

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

Rewrite using the commutative property of multiplication.

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

Combine $\cos (θ)$ and $\frac{1}{\sin (θ)}$ .

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

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

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

Rewrite using the commutative property of multiplication.

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Add $\frac{1}{\sin (θ)}$ and $0$ .

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

Combine the numerators over the common denominator.

$\frac{1 - \cos^{2} (θ)}{\sin (θ)}$

Apply pythagorean identity.

$\frac{\sin^{2} (θ)}{\sin (θ)}$

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

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

$\frac{\sin (θ) \sin (θ)}{\sin (θ)}$

Cancel the common factors.

Multiply by $1$ .

$\frac{\sin (θ) \sin (θ)}{\sin (θ) \cdot 1}$

Cancel the common factor.

$\frac{\sin (θ) \sin (θ)}{\sin (θ) \cdot 1}$

Rewrite the expression.

$\frac{\sin (θ)}{1}$

Divide $\sin (θ)$ by $1$ .

$\sin (θ)$

Simplify (csc(theta)+cot(theta))(1-cos(theta))

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