Trigonometry Examples

$\frac{\csc (a) + 1}{\csc (a) - 1} = \frac{1 + \sin (a)}{1 - \sin (a)}$

Step 1

Start on the left side.

$\frac{\csc (a) + 1}{\csc (a) - 1}$

Step 2

Convert to sines and cosines.

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Step 2.1

Apply the reciprocal identity to $\csc (a)$ .

$\frac{\frac{1}{\sin (a)} + 1}{\csc (a) - 1}$

Step 2.2

Apply the reciprocal identity to $\csc (a)$ .

$\frac{\frac{1}{\sin (a)} + 1}{\frac{1}{\sin (a)} - 1}$

Step 3

Simplify.

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Step 3.1

Multiply the numerator and denominator of the fraction by $\sin (a)$ .

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Step 3.1.1

Multiply $\frac{\frac{1}{\sin (a)} + 1}{\frac{1}{\sin (a)} - 1}$ by $\frac{\sin (a)}{\sin (a)}$ .

$\frac{\sin (a)}{\sin (a)} \cdot \frac{\frac{1}{\sin (a)} + 1}{\frac{1}{\sin (a)} - 1}$

Step 3.1.2

Combine.

$\frac{\sin (a) (\frac{1}{\sin (a)} + 1)}{\sin (a) (\frac{1}{\sin (a)} - 1)}$

Step 3.2

Apply the distributive property.

$\frac{\sin (a) \frac{1}{\sin (a)} + \sin (a) \cdot 1}{\sin (a) \frac{1}{\sin (a)} + \sin (a) \cdot - 1}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

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

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Step 3.3.1.1

Cancel the common factor.

$\frac{\sin (a) \frac{1}{\sin (a)} + \sin (a) \cdot 1}{\sin (a) \frac{1}{\sin (a)} + \sin (a) \cdot - 1}$

Step 3.3.1.2

Rewrite the expression.

$\frac{1 + \sin (a) \cdot 1}{\sin (a) \frac{1}{\sin (a)} + \sin (a) \cdot - 1}$

Step 3.3.2

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

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Step 3.3.2.1

Cancel the common factor.

$\frac{1 + \sin (a) \cdot 1}{\sin (a) \frac{1}{\sin (a)} + \sin (a) \cdot - 1}$

Step 3.3.2.2

Rewrite the expression.

$\frac{1 + \sin (a) \cdot 1}{1 + \sin (a) \cdot - 1}$

Step 3.4

Multiply $\sin (a)$ by $1$ .

$\frac{1 + \sin (a)}{1 + \sin (a) \cdot - 1}$

Step 3.5

Simplify the denominator.

$\frac{1 + \sin (a)}{1 - \sin (a)}$

Step 4

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

$\frac{\csc (a) + 1}{\csc (a) - 1} = \frac{1 + \sin (a)}{1 - \sin (a)}$ is an identity