Trigonometry Examples

$\csc (θ) = \frac{1}{\sin (θ)}$

Step 1

Multiply both sides by $\sin (θ)$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 2.1.1.1

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

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

Step 2.1.1.2

Cancel the common factors.

$1 = \frac{1}{\sin (θ)} \sin (θ)$

Step 2.2

Simplify the right side.

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Step 2.2.1

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

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Step 2.2.1.1

Cancel the common factor.

$1 = \frac{1}{\sin (θ)} \sin (θ)$

Step 2.2.1.2

Rewrite the expression.

$1 = 1$

Step 3

Since $1 = 1$ , the equation will always be true for any value of $θ$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$