Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

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

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

Step 6.3

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

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Step 6.3.1

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

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

Step 6.3.2

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

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

Step 6.3.3

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

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

Step 6.3.4

Add $1$ and $1$ .

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

Step 6.4

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

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Step 6.4.1

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

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

Step 6.4.2

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

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

Step 6.4.3

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

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

Step 6.4.4

Add $1$ and $1$ .

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

Step 6.5

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

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Step 6.5.1

Rearrange terms.

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

Step 6.5.2

Apply pythagorean identity.

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

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

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

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Step 7.1.1

Cancel the common factor.

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

Step 7.1.2

Rewrite the expression.

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

Step 7.2

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

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Step 7.2.1

Reorder terms.

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

$1$