Trigonometry Examples

$\frac{1 + \tan (x)}{1 - \tan (x)} + \frac{1 + \cot (x)}{1 - \cot (x)} = 0$

Step 1

Simplify $\frac{1 + \tan (x)}{1 - \tan (x)} + \frac{1 + \cot (x)}{1 - \cot (x)}$ .

Tap for more steps...

Step 1.1

To write $\frac{1 + \tan (x)}{1 - \tan (x)}$ as a fraction with a common denominator, multiply by $\frac{1 - \cot (x)}{1 - \cot (x)}$ .

$\frac{1 + \tan (x)}{1 - \tan (x)} \cdot \frac{1 - \cot (x)}{1 - \cot (x)} + \frac{1 + \cot (x)}{1 - \cot (x)} = 0$

Step 1.2

To write $\frac{1 + \cot (x)}{1 - \cot (x)}$ as a fraction with a common denominator, multiply by $\frac{1 - \tan (x)}{1 - \tan (x)}$ .

$\frac{1 + \tan (x)}{1 - \tan (x)} \cdot \frac{1 - \cot (x)}{1 - \cot (x)} + \frac{1 + \cot (x)}{1 - \cot (x)} \cdot \frac{1 - \tan (x)}{1 - \tan (x)} = 0$

Step 1.3

Write each expression with a common denominator of $(1 - \tan (x)) (1 - \cot (x))$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 1.3.1

Multiply $\frac{1 + \tan (x)}{1 - \tan (x)}$ by $\frac{1 - \cot (x)}{1 - \cot (x)}$ .

$\frac{(1 + \tan (x)) (1 - \cot (x))}{(1 - \tan (x)) (1 - \cot (x))} + \frac{1 + \cot (x)}{1 - \cot (x)} \cdot \frac{1 - \tan (x)}{1 - \tan (x)} = 0$

Step 1.3.2

Multiply $\frac{1 + \cot (x)}{1 - \cot (x)}$ by $\frac{1 - \tan (x)}{1 - \tan (x)}$ .

$\frac{(1 + \tan (x)) (1 - \cot (x))}{(1 - \tan (x)) (1 - \cot (x))} + \frac{(1 + \cot (x)) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.3.3

Reorder the factors of $(1 - \tan (x)) (1 - \cot (x))$ .

$\frac{(1 + \tan (x)) (1 - \cot (x))}{(1 - \cot (x)) (1 - \tan (x))} + \frac{(1 + \cot (x)) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.4

Combine the numerators over the common denominator.

$\frac{(1 + \tan (x)) (1 - \cot (x)) + (1 + \cot (x)) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5

Simplify the numerator.

Tap for more steps...

Step 1.5.1

Expand $(1 + \tan (x)) (1 - \cot (x))$ using the FOIL Method.

Tap for more steps...

Step 1.5.1.1

Apply the distributive property.

$\frac{1 (1 - \cot (x)) + \tan (x) (1 - \cot (x)) + (1 + \cot (x)) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.1.2

Apply the distributive property.

$\frac{1 \cdot 1 + 1 (- \cot (x)) + \tan (x) (1 - \cot (x)) + (1 + \cot (x)) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.1.3

Apply the distributive property.

$\frac{1 \cdot 1 + 1 (- \cot (x)) + \tan (x) \cdot 1 + \tan (x) (- \cot (x)) + (1 + \cot (x)) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.2

Simplify and combine like terms.

Tap for more steps...

Step 1.5.2.1

Simplify each term.

Tap for more steps...

Step 1.5.2.1.1

Multiply $1$ by $1$ .

$\frac{1 + 1 (- \cot (x)) + \tan (x) \cdot 1 + \tan (x) (- \cot (x)) + (1 + \cot (x)) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.2.1.2

Multiply $- \cot (x)$ by $1$ .

$\frac{1 - \cot (x) + \tan (x) \cdot 1 + \tan (x) (- \cot (x)) + (1 + \cot (x)) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.2.1.3

Multiply $\tan (x)$ by $1$ .

$\frac{1 - \cot (x) + \tan (x) + \tan (x) (- \cot (x)) + (1 + \cot (x)) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.2.1.4

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

Tap for more steps...

Step 1.5.2.1.4.1

Reorder $\tan (x)$ and $- \cot (x)$ .

$\frac{1 - \cot (x) + \tan (x) - \cot (x) \tan (x) + (1 + \cot (x)) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.2.1.4.2

Add parentheses.

$\frac{1 - \cot (x) + \tan (x) - (\cot (x) \tan (x)) + (1 + \cot (x)) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.2.1.4.3

Rewrite $\tan (x) (- \cot (x))$ in terms of sines and cosines.

$\frac{1 - \cot (x) + \tan (x) - (\frac{\cos (x)}{\sin (x)} \cdot \frac{\sin (x)}{\cos (x)}) + (1 + \cot (x)) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.2.1.4.4

Cancel the common factors.

$\frac{1 - \cot (x) + \tan (x) - 1 \cdot 1 + (1 + \cot (x)) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.2.1.5

Multiply $- 1$ by $1$ .

$\frac{1 - \cot (x) + \tan (x) - 1 + (1 + \cot (x)) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.2.2

Subtract $1$ from $1$ .

$\frac{0 - \cot (x) + \tan (x) + (1 + \cot (x)) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.2.3

Subtract $\cot (x)$ from $0$ .

$\frac{- \cot (x) + \tan (x) + (1 + \cot (x)) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.3

Expand $(1 + \cot (x)) (1 - \tan (x))$ using the FOIL Method.

Tap for more steps...

Step 1.5.3.1

Apply the distributive property.

$\frac{- \cot (x) + \tan (x) + 1 (1 - \tan (x)) + \cot (x) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.3.2

Apply the distributive property.

$\frac{- \cot (x) + \tan (x) + 1 \cdot 1 + 1 (- \tan (x)) + \cot (x) (1 - \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.3.3

Apply the distributive property.

$\frac{- \cot (x) + \tan (x) + 1 \cdot 1 + 1 (- \tan (x)) + \cot (x) \cdot 1 + \cot (x) (- \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.4

Simplify and combine like terms.

Tap for more steps...

Step 1.5.4.1

Simplify each term.

Tap for more steps...

Step 1.5.4.1.1

Multiply $1$ by $1$ .

$\frac{- \cot (x) + \tan (x) + 1 + 1 (- \tan (x)) + \cot (x) \cdot 1 + \cot (x) (- \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.4.1.2

Multiply $- \tan (x)$ by $1$ .

$\frac{- \cot (x) + \tan (x) + 1 - \tan (x) + \cot (x) \cdot 1 + \cot (x) (- \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.4.1.3

Multiply $\cot (x)$ by $1$ .

$\frac{- \cot (x) + \tan (x) + 1 - \tan (x) + \cot (x) + \cot (x) (- \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.4.1.4

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

Tap for more steps...

Step 1.5.4.1.4.1

Reorder $\cot (x)$ and $- \tan (x)$ .

$\frac{- \cot (x) + \tan (x) + 1 - \tan (x) + \cot (x) - \tan (x) \cot (x)}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.4.1.4.2

Add parentheses.

$\frac{- \cot (x) + \tan (x) + 1 - \tan (x) + \cot (x) - (\tan (x) \cot (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.4.1.4.3

Reorder $\tan (x)$ and $\cot (x)$ .

$\frac{- \cot (x) + \tan (x) + 1 - \tan (x) + \cot (x) - (\cot (x) \tan (x))}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.4.1.4.4

Rewrite $\cot (x) (- \tan (x))$ in terms of sines and cosines.

$\frac{- \cot (x) + \tan (x) + 1 - \tan (x) + \cot (x) - (\frac{\cos (x)}{\sin (x)} \cdot \frac{\sin (x)}{\cos (x)})}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.4.1.4.5

Cancel the common factors.

$\frac{- \cot (x) + \tan (x) + 1 - \tan (x) + \cot (x) - 1 \cdot 1}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.4.1.5

Multiply $- 1$ by $1$ .

$\frac{- \cot (x) + \tan (x) + 1 - \tan (x) + \cot (x) - 1}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.4.2

Subtract $1$ from $1$ .

$\frac{- \cot (x) + \tan (x) + 0 - \tan (x) + \cot (x)}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.4.3

Subtract $\tan (x)$ from $0$ .

$\frac{- \cot (x) + \tan (x) - \tan (x) + \cot (x)}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.5

Add $- \cot (x)$ and $\cot (x)$ .

$\frac{0 + \tan (x) - \tan (x)}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.6

Add $0$ and $\tan (x)$ .

$\frac{\tan (x) - \tan (x)}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.5.7

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

$\frac{0}{(1 - \cot (x)) (1 - \tan (x))} = 0$

Step 1.6

Divide $0$ by $(1 - \cot (x)) (1 - \tan (x))$ .

$0 = 0$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 0.1, 0.1, - \frac{1}{5}, \frac{1}{5}, \frac{2}{5}, - \frac{1}{2}, \frac{1}{2}$

Step 3