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

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

Simplify each term.

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

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

Rewrite using the commutative property of multiplication.

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

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

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

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

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

Rewrite using the commutative property of multiplication.

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Add $\frac{1}{\sin (x)}$ and $0$ .

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

Combine the numerators over the common denominator.

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

Apply pythagorean identity.

$\frac{\sin^{2} (x)}{\sin (x)}$

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

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

$\frac{\sin (x) \sin (x)}{\sin (x)}$

Cancel the common factors.

Multiply by $1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

$\sin (x)$

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

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