Trigonometry Examples

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

Start on the right side.

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

Apply the tangent double-angle identity.

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

Convert to sines and cosines.

Tap for more steps...

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

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

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

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

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

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

Simplify.

Tap for more steps...

Multiply the numerator by the reciprocal of the denominator.

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

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

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

Simplify the numerator.

Tap for more steps...

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

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

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

Combine the numerators over the common denominator.

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

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

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

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

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

Simplify the denominator.

Tap for more steps...

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Combine.

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

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

Tap for more steps...

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

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

Cancel the common factors.

Tap for more steps...

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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