Trigonometry Examples

$\tan (x + \frac{π}{2}) = - \cot (x)$

Start on the left side.

$\tan (x + \frac{π}{2})$

Write $\tan (x + \frac{π}{2})$ in sines and cosines using the quotient identity.

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

Apply the sum of angles identity.

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

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

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

Simplify the expression.

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Simplify the numerator.

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The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\frac{\sin (x) \cdot 0 + \cos (x) \sin (\frac{π}{2})}{\cos (x) \cos (\frac{π}{2}) - \sin (x) \sin (\frac{π}{2})}$

Multiply $\sin (x)$ by $0$ .

$\frac{0 + \cos (x) \sin (\frac{π}{2})}{\cos (x) \cos (\frac{π}{2}) - \sin (x) \sin (\frac{π}{2})}$

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{0 + \cos (x) \cdot 1}{\cos (x) \cos (\frac{π}{2}) - \sin (x) \sin (\frac{π}{2})}$

Multiply $\cos (x)$ by $1$ .

$\frac{0 + \cos (x)}{\cos (x) \cos (\frac{π}{2}) - \sin (x) \sin (\frac{π}{2})}$

Add $0$ and $\cos (x)$ .

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

Simplify the denominator.

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The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\frac{\cos (x)}{\cos (x) \cdot 0 - \sin (x) \sin (\frac{π}{2})}$

Multiply $\cos (x)$ by $0$ .

$\frac{\cos (x)}{0 - \sin (x) \sin (\frac{π}{2})}$

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{\cos (x)}{0 - \sin (x) \cdot 1}$

Multiply $- 1$ by $1$ .

$\frac{\cos (x)}{0 - \sin (x)}$

Subtract $\sin (x)$ from $0$ .

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

Separate fractions.

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

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\frac{1}{- 1} \cot (x)$

Divide $1$ by $- 1$ .

$- \cot (x)$

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

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