Algebra Examples

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

Step 1

Start on the right side.

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

Step 2

Apply the tangent double-angle identity.

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

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.

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Simplify.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

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

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

Step 4.3

Simplify the numerator.

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

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 $b = \frac{\sin (x)}{\cos (x)}$ .

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

Step 4.3.4

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

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

Step 4.3.5

Combine the numerators over the common denominator.

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

Step 4.3.6

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

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

Step 4.3.7

Combine the numerators over the common denominator.

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

Simplify the denominator.

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Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

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

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

Step 4.6.4

Add $1$ and $1$ .

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

Step 4.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.8

Combine.

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

Step 4.9

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

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Step 4.9.1

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

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

Step 4.9.2

Cancel the common factors.

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Step 4.9.2.1

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

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

Step 4.9.2.2

Cancel the common factor.

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

Step 4.9.2.3

Rewrite the expression.

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

Step 4.10

Move $2$ to the left of $\cos (x)$ .

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

Step 5

Rewrite $\frac{(\cos (x) + \sin (x)) (\cos (x) - \sin (x))}{2 \cos (x) \sin (x)}$ as $\frac{(1 - \tan^{2} (x))}{2 \tan (x)}$ .

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

Step 6

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

$\frac{1 - \tan^{2} (x)}{2 \tan (x)} = \frac{1}{\tan (2 x)}$ is an identity