Trigonometry Examples

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

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

Step 1

Start on the left side.

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

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

Step 2

Convert to sines and cosines.

Tap for more steps...

Step 2.1

Apply the reciprocal identity to

\csc (a)

$\csc (a)$ .

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

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

Step 2.2

Apply the reciprocal identity to

\csc (a)

$\csc (a)$ .

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

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

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

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Multiply the numerator and denominator of the fraction by

\sin (a)

$\sin (a)$ .

Tap for more steps...

Step 3.1.1

Multiply

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

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

\frac{\sin (a)}{\sin (a)}

$\frac{\sin (a)}{\sin (a)}$ .

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

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

Step 3.1.2

Combine.

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

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

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

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

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}

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

Step 3.3

Simplify by cancelling.

Tap for more steps...

Step 3.3.1

Cancel the common factor of

\sin (a)

$\sin (a)$ .

Tap for more steps...

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}

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

Step 3.3.1.2

Rewrite the expression.

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

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

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

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

Step 3.3.2

Cancel the common factor of

\sin (a)

$\sin (a)$ .

Tap for more steps...

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}

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

Step 3.3.2.2

Rewrite the expression.

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

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

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

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

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

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

Step 3.4

Multiply

\sin (a)

$\sin (a)$ by

1

$1$ .

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

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

Step 3.5

Simplify the denominator.

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

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

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

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

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)}

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