Trigonometry Examples

$\frac{1 + \csc (x)}{1 - \csc (x)} + \frac{1 + \sin (x)}{1 - \sin (x)} = 0$

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify each term.

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

Step 4

Add fractions.

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Step 4.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 4.2

Factor $- 1$ out of $- \sin (x)$ .

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

Step 4.3

Factor $- 1$ out of $- 1 (- 1) - (\sin (x))$ .

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

Step 4.4

Move a negative from the denominator of $\frac{1 + \sin (x)}{- 1 (- 1 + \sin (x))}$ to the numerator.

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

Step 4.5

Reorder terms.

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

Step 4.6

Combine the numerators over the common denominator.

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

Step 5

Simplify numerator.

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Step 5.1

Simplify each term.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Multiply $- 1$ by $1$ .

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

Step 5.2

Subtract $\sin (x)$ from $\sin (x)$ .

$\frac{0 + 1 - 1}{- 1 + \sin (x)}$

Step 5.3

Add $0$ and $1$ .

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

Step 5.4

Subtract $1$ from $1$ .

$\frac{0}{- 1 + \sin (x)}$

Step 6

Divide $0$ by $- 1 + \sin (x)$ .

$0$

Step 7

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

$\frac{1 + \csc (x)}{1 - \csc (x)} + \frac{1 + \sin (x)}{1 - \sin (x)} = 0$ is an identity