Calculus Examples

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

Step 1

Start on the left side.

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

Step 2

Write $\cot (x)$ in sines and cosines using the quotient identity.

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

Step 3

Simplify.

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Step 3.1

Multiply the numerator and denominator of the fraction by $\sin (x)$ .

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Step 3.1.1

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

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

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

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Multiply $\sin (x) \sin (x)$ .

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Step 3.4.1.1

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

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

Step 3.4.1.2

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

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

Step 3.4.1.3

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

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

Step 3.4.1.4

Add $1$ and $1$ .

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

Step 3.4.2

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

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Step 3.4.2.1

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

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

Step 3.4.2.2

Factor $\sin (x)$ out of $\sin (x) \cos (x)$ .

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

Step 3.4.2.3

Factor $\sin (x)$ out of $\sin (x) \sin (x) + \sin (x) (\cos (x))$ .

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

Step 3.5

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

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

Step 3.6

Cancel the common factor of $\sin (x) + \cos (x)$ .

$\sin (x)$

Step 4

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{\sin (x) + \cos (x)}{1 + \cot (x)} = \sin (x)$ is an identity