Trigonometry Examples

$\cot (θ) = \cos (θ) \csc (θ)$

Step 1

Simplify the left side.

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Step 1.1

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

$\frac{\cos (θ)}{\sin (θ)} = \cos (θ) \csc (θ)$

Step 2

Simplify the right side.

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Step 2.1

Simplify $\cos (θ) \csc (θ)$ .

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Step 2.1.1

Rewrite $\csc (θ)$ in terms of sines and cosines.

$\frac{\cos (θ)}{\sin (θ)} = \cos (θ) \frac{1}{\sin (θ)}$

Step 2.1.2

Combine $\cos (θ)$ and $\frac{1}{\sin (θ)}$ .

$\frac{\cos (θ)}{\sin (θ)} = \frac{\cos (θ)}{\sin (θ)}$

Step 3

Multiply both sides of the equation by $\sin (θ)$ .

$\sin (θ) \frac{\cos (θ)}{\sin (θ)} = \sin (θ) \frac{\cos (θ)}{\sin (θ)}$

Step 4

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

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Step 4.1

Cancel the common factor.

$\sin (θ) \frac{\cos (θ)}{\sin (θ)} = \sin (θ) \frac{\cos (θ)}{\sin (θ)}$

Step 4.2

Rewrite the expression.

$\cos (θ) = \sin (θ) \frac{\cos (θ)}{\sin (θ)}$

Step 5

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

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Step 5.1

Cancel the common factor.

$\cos (θ) = \sin (θ) \frac{\cos (θ)}{\sin (θ)}$

Step 5.2

Rewrite the expression.

$\cos (θ) = \cos (θ)$

Step 6

For the two functions to be equal, the arguments of each must be equal.

$θ = θ$

Step 7

Move all terms containing $θ$ to the left side of the equation.

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Step 7.1

Subtract $θ$ from both sides of the equation.

$θ - θ = 0$

Step 7.2

Subtract $θ$ from $θ$ .

$0 = 0$

Step 8

Since $0 = 0$ , the equation will always be true for any value of $θ$ .

All real numbers

Step 9

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$