Precalculus Examples

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

Step 1

Start on the right side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

Simplify the denominator.

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Step 3.1.1.1

Write $1$ as a fraction with a common denominator.

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

Step 3.1.1.2

Combine the numerators over the common denominator.

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

Step 3.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.3

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

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Step 3.1.3.1

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

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

Step 3.1.3.2

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

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

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

Step 3.1.3.5

Add $1$ and $1$ .

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

Step 3.1.4

Simplify the denominator.

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Step 3.1.4.1

Write $1$ as a fraction with a common denominator.

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

Step 3.1.4.2

Combine the numerators over the common denominator.

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

Step 3.1.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.6

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

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Step 3.1.6.1

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

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

Step 3.1.6.2

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

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

Step 3.1.6.3

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

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

Step 3.1.6.4

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

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

Step 3.1.6.5

Add $1$ and $1$ .

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

Step 3.2

Factor $- 1$ out of $\cos (x)$ .

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

Step 3.3

Factor $- 1$ out of $- \sin (x)$ .

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

Step 3.4

Factor $- 1$ out of $- 1 (- \cos (x)) - (\sin (x))$ .

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

Step 3.5

Move a negative from the denominator of $\frac{{\cos (x)}^{2}}{- 1 (- \cos (x) + \sin (x))}$ to the numerator.

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

Step 3.6

Reorder terms.

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

Step 3.7

Combine the numerators over the common denominator.

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

Step 3.8

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \sin (x)$ and $b = \cos (x)$ .

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

Step 3.9

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

$\sin (x) + \cos (x)$

Step 4

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

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