Precalculus Examples

$\frac{2 (\tan (x) - \cot (x))}{\tan^{2} (x) - \cot^{2} (x)} = \sin (2 x)$

Step 1

Start on the left side.

$\frac{2 (\tan (x) - \cot (x))}{\tan^{2} (x) - \cot^{2} (x)}$

Step 2

Simplify the expression.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

Simplify the denominator.

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Step 2.2.1

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

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.2.4

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

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

Step 2.3

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

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Step 2.3.1

Cancel the common factor.

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

Step 2.3.2

Rewrite the expression.

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

Step 3

Simplify.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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Step 3.1.3.1

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

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

Step 3.1.3.2

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

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

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

Step 3.1.4

Combine the numerators over the common denominator.

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

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

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

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Step 3.1.5.1.1

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

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

Step 3.1.5.1.2

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

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

Step 3.1.5.1.3

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

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

Step 3.1.5.1.4

Add $1$ and $1$ .

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

Step 3.1.5.2

Multiply $\cos (x) \cos (x)$ .

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Step 3.1.5.2.1

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

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

Step 3.1.5.2.2

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

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

Step 3.1.5.2.3

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

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

Step 3.1.5.2.4

Add $1$ and $1$ .

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

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3

Combine $2$ and $\frac{\cos (x) \sin (x)}{{\sin (x)}^{2} + {\cos (x)}^{2}}$ .

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

Step 4

Simplify the expression.

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Step 4.1

Apply pythagorean identity.

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

Step 4.2

Divide $2 \cos (x) \sin (x)$ by $1$ .

$2 \cos (x) \sin (x)$

Step 4.3

Reorder $2 \cos (x)$ and $\sin (x)$ .

$\sin (x) (2 \cos (x))$

Step 4.4

Reorder $\sin (x)$ and $2$ .

$2 \cdot \sin (x) \cos (x)$

Step 4.5

Apply the sine double-angle identity.

$\sin (2 x)$

Step 5

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

$\frac{2 (\tan (x) - \cot (x))}{\tan^{2} (x) - \cot^{2} (x)} = \sin (2 x)$ is an identity