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 difference 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 difference 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 $- 1$ by $0$ .

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

Step 5.1.5

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

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

Step 5.1.6

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 $1$ .

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

Step 5.2.4

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

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

Step 5.2.5

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

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

Step 5.3

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

$\cot (x)$

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