Trigonometry Examples

$\tan (\frac{π}{2} + x) = - \cot (x)$

Step 1

Start on the left side.

$\tan (\frac{π}{2} + x)$

Step 2

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

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

Step 3

Apply the sum of angles identity.

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

Step 4

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

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

Step 5

Simplify the expression.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

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

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

Step 5.1.5

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

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

Step 5.2

Simplify the denominator.

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

Multiply $- 1$ by $1$ .

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

Step 5.2.5

Rewrite $- 1 \sin (x)$ as $- \sin (x)$ .

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

Step 5.2.6

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

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

Step 5.3

Separate fractions.

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

Step 5.4

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

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

Step 5.5

Divide $1$ by $- 1$ .

$- \cot (x)$

Step 6

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

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