Trigonometry Examples

$\cot (x - \frac{π}{2}) = - \tan (x)$

Step 1

Start on the left side.

$\cot (x - \frac{π}{2})$

Step 2

Apply the difference of angles identity.

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

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

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

Step 3.1.2

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

$\frac{0 + 1}{\cot (\frac{π}{2}) - \cot (x)}$

Step 3.1.3

Add $0$ and $1$ .

$\frac{1}{\cot (\frac{π}{2}) - \cot (x)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

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

$\frac{1}{0 - \cot (x)}$

Step 3.2.2

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

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

Step 3.3

Cancel the common factor of $1$ and $- 1$ .

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Step 3.3.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 3.3.2

Move the negative in front of the fraction.

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

Step 3.4

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

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

Step 3.5

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (x)}{\sin (x)}$ .

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

Step 3.6

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

$- \tan (x)$

Step 4

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

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