Trigonometry Examples

$\cot (\frac{π}{2} - x) = \tan (x)$

Step 1

Start on the left side.

$\cot (\frac{π}{2} - x)$

Step 2

Apply the difference of angles identity.

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

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 $0$ .

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

Step 3.1.2

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

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

Step 3.1.3

Add $0$ and $1$ .

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

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

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

Step 3.2.2

Multiply $- 1$ by $0$ .

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

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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.

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