Trigonometry Examples

$\tan (\frac{π}{2} - x) \tan (x) = 1$

Step 1

Start on the left side.

$\tan (\frac{π}{2} - x) \tan (x)$

Step 2

Simplify the expression.

Tap for more steps...

Step 2.1

Rewrite $\tan (\frac{π}{2} - x)$ in terms of sines and cosines.

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

Step 2.2

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 2.3

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

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

Step 3

Apply the difference of angles identity.

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

Step 4

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

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

Step 5

Simplify the expression.

Tap for more steps...

Step 5.1

Simplify the numerator.

Tap for more steps...

Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

Multiply $- 1$ by $0$ .

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

Step 5.1.5

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

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

Step 5.1.6

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

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

Step 5.2

Simplify the denominator.

Tap for more steps...

Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

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

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

Step 5.2.5

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

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

Step 5.3

Cancel the common factor of $\cos (x)$ .

Tap for more steps...

Step 5.3.1

Cancel the common factor.

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

Step 5.3.2

Rewrite the expression.

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

Step 5.4

Cancel the common factor of $\sin (x)$ .

Tap for more steps...

Step 5.4.1

Cancel the common factor.

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

Step 5.4.2

Rewrite the expression.

$1$

Step 6

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

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