Simplify sin(a)(cot(a)+(sin(a))/(1+cos(a)))

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $\cos (a)$ by $1$ .

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

Multiply $\cos (a) \cos (a)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $\sin (a) \sin (a)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

Rearrange terms.

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

Apply pythagorean identity.

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

Reduce the expression by cancelling the common factors.

Cancel the common factor of $\sin (a)$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{\cos (a) + 1}{1 + \cos (a)}$

Cancel the common factor of $\cos (a) + 1$ and $1 + \cos (a)$ .

Reorder terms.

$\frac{1 + \cos (a)}{1 + \cos (a)}$

Cancel the common factor.

$\frac{1 + \cos (a)}{1 + \cos (a)}$

Rewrite the expression.

$1$

Simplify sin(a)(cot(a)+(sin(a))/(1+cos(a)))

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