Calculus Examples

$\cot (\frac{π}{2} - θ) \cot (θ) = 1$

Step 1

Start on the left side.

$\cot (\frac{π}{2} - θ) \cot (θ)$

Step 2

Apply the difference of angles identity.

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

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 $0$ .

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

Step 3.1.2

Multiply $0$ by $\cot (θ)$ .

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

Step 3.1.3

Add $0$ and $1$ .

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

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

$\frac{1}{\cot (θ) - 0} \cot (θ)$

Step 3.2.2

Multiply $- 1$ by $0$ .

$\frac{1}{\cot (θ) + 0} \cot (θ)$

Step 3.2.3

Add $\cot (θ)$ and $0$ .

$\frac{1}{\cot (θ)} \cot (θ)$

Step 3.3

Cancel the common factor of $\cot (θ)$ .

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Step 3.3.1

Cancel the common factor.

$\frac{1}{\cot (θ)} \cot (θ)$

Step 3.3.2

Rewrite the expression.

$1$

Step 4

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

$\cot (\frac{π}{2} - θ) \cot (θ) = 1$ is an identity