Trigonometry Examples

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

Step 1

Start on the left side.

$\tan (2 x)$

Step 2

Apply the tangent double-angle identity.

$\frac{2 \tan (x)}{1 - \tan^{2} (x)}$

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Simplify the denominator.

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Step 4.2.1

Rewrite $1$ as $1^{2}$ .

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

Step 4.2.2

Rewrite $\frac{{\sin (x)}^{2}}{{\cos (x)}^{2}}$ as ${(\frac{\sin (x)}{\cos (x)})}^{2}$ .

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

Step 4.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{\sin (x)}{\cos (x)}$ .

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

Step 4.2.4

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

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

Step 4.2.5

Combine the numerators over the common denominator.

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

Step 4.2.6

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

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

Step 4.2.7

Combine the numerators over the common denominator.

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

Step 4.3

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

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

Step 4.4

Simplify the denominator.

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Step 4.4.1

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

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

Step 4.4.2

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

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

Step 4.4.3

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

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

Step 4.4.4

Add $1$ and $1$ .

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

Step 4.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.6

Combine.

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

Step 4.7

Cancel the common factor of ${\cos (x)}^{2}$ and $\cos (x)$ .

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

Step 5

Reorder terms.

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

Step 6

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

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

Step 7

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

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