Trigonometry Examples

$\csc (θ) = \frac{1}{\sin (θ)}$

Step 1

Multiply each term by a factor of $1$ that will equate all the denominators. In this case, all terms need a denominator of $\sin (θ)$ .

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

Step 2

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $\sin (θ)$ .

$\csc (θ) \cdot \sin (θ)$

Step 3

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

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Step 3.1

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

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

Step 3.2

Cancel the common factors.

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

Step 4

Simplify the left side.

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Step 4.1

Simplify $\csc (θ) \cdot \frac{\sin (θ)}{\sin (θ)}$ .

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Step 4.1.1

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

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Step 4.1.1.1

Cancel the common factor.

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

Step 4.1.1.2

Rewrite the expression.

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

Step 4.1.2

Multiply $\csc (θ)$ by $1$ .

$\csc (θ) = \frac{1}{\sin (θ)}$

Step 4.1.3

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

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

Step 5

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

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

Step 6

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

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$1 = \sin (θ) \frac{1}{\sin (θ)}$

Step 7

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

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Step 7.1

Cancel the common factor.

$1 = \sin (θ) \frac{1}{\sin (θ)}$

Step 7.2

Rewrite the expression.

$1 = 1$

Step 8

Since $1 = 1$ , the equation will always be true.

Always true