Trigonometry Examples

$\cot (x) + \tan (x) = \frac{2}{\sin (2 x)}$

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

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

$\frac{\cos (x)}{\sin (x)} + \tan (x) = \frac{2}{\sin (2 x)}$

Step 1.1.2

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

$\frac{\cos (x)}{\sin (x)} + \frac{\sin (x)}{\cos (x)} = \frac{2}{\sin (2 x)}$

Step 2

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

$\sin (x) (\frac{\cos (x)}{\sin (x)} + \frac{\sin (x)}{\cos (x)}) = \sin (x) \frac{2}{\sin (2 x)}$

Step 3

Apply the distributive property.

$\sin (x) \frac{\cos (x)}{\sin (x)} + \sin (x) \frac{\sin (x)}{\cos (x)} = \sin (x) \frac{2}{\sin (2 x)}$

Step 4

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

Tap for more steps...

Step 4.1

Cancel the common factor.

$\sin (x) \frac{\cos (x)}{\sin (x)} + \sin (x) \frac{\sin (x)}{\cos (x)} = \sin (x) \frac{2}{\sin (2 x)}$

Step 4.2

Rewrite the expression.

$\cos (x) + \sin (x) \frac{\sin (x)}{\cos (x)} = \sin (x) \frac{2}{\sin (2 x)}$

Step 5

Multiply $\sin (x) \frac{\sin (x)}{\cos (x)}$ .

Tap for more steps...

Step 5.1

Combine $\sin (x)$ and $\frac{\sin (x)}{\cos (x)}$ .

$\cos (x) + \frac{\sin (x) \sin (x)}{\cos (x)} = \sin (x) \frac{2}{\sin (2 x)}$

Step 5.2

Raise $\sin (x)$ to the power of $1$ .

$\cos (x) + \frac{\sin^{1} (x) \sin (x)}{\cos (x)} = \sin (x) \frac{2}{\sin (2 x)}$

Step 5.3

Raise $\sin (x)$ to the power of $1$ .

$\cos (x) + \frac{\sin^{1} (x) \sin^{1} (x)}{\cos (x)} = \sin (x) \frac{2}{\sin (2 x)}$

Step 5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cos (x) + \frac{{\sin (x)}^{1 + 1}}{\cos (x)} = \sin (x) \frac{2}{\sin (2 x)}$

Step 5.5

Add $1$ and $1$ .

$\cos (x) + \frac{\sin^{2} (x)}{\cos (x)} = \sin (x) \frac{2}{\sin (2 x)}$

Step 6

Combine $\sin (x)$ and $\frac{2}{\sin (2 x)}$ .

$\cos (x) + \frac{\sin^{2} (x)}{\cos (x)} = \frac{\sin (x) \cdot 2}{\sin (2 x)}$

Step 7

Apply the sine double-angle identity.

$\cos (x) + \frac{\sin^{2} (x)}{\cos (x)} = \frac{\sin (x) \cdot 2}{2 \sin (x) \cos (x)}$

Step 8

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

Tap for more steps...

Step 8.1

Cancel the common factor.

$\cos (x) + \frac{\sin^{2} (x)}{\cos (x)} = \frac{\sin (x) \cdot 2}{2 \sin (x) \cos (x)}$

Step 8.2

Rewrite the expression.

$\cos (x) + \frac{\sin^{2} (x)}{\cos (x)} = \frac{2}{2 \cos (x)}$

Step 9

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.1

Cancel the common factor.

$\cos (x) + \frac{\sin^{2} (x)}{\cos (x)} = \frac{2}{2 \cos (x)}$

Step 9.2

Rewrite the expression.

$\cos (x) + \frac{\sin^{2} (x)}{\cos (x)} = \frac{1}{\cos (x)}$

Step 10

Subtract $\frac{1}{\cos (x)}$ from both sides of the equation.

$\cos (x) + \frac{\sin^{2} (x)}{\cos (x)} - \frac{1}{\cos (x)} = 0$

Step 11

Simplify $\cos (x) + \frac{\sin^{2} (x)}{\cos (x)} - \frac{1}{\cos (x)}$ .

Tap for more steps...

Step 11.1

Combine the numerators over the common denominator.

$\cos (x) + \frac{\sin^{2} (x) - 1}{\cos (x)} = 0$

Step 11.2

Reorder $\sin^{2} (x)$ and $- 1$ .

$\cos (x) + \frac{- 1 + \sin^{2} (x)}{\cos (x)} = 0$

Step 11.3

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

$\cos (x) + \frac{- 1 (1) + \sin^{2} (x)}{\cos (x)} = 0$

Step 11.4

Factor $- 1$ out of $\sin^{2} (x)$ .

$\cos (x) + \frac{- 1 (1) - 1 (- \sin^{2} (x))}{\cos (x)} = 0$

Step 11.5

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (x))$ .

$\cos (x) + \frac{- 1 (1 - \sin^{2} (x))}{\cos (x)} = 0$

Step 11.6

Rewrite $- 1 (1 - \sin^{2} (x))$ as $- (1 - \sin^{2} (x))$ .

$\cos (x) + \frac{- (1 - \sin^{2} (x))}{\cos (x)} = 0$

Step 11.7

Apply pythagorean identity.

$\cos (x) + \frac{- \cos^{2} (x)}{\cos (x)} = 0$

Step 11.8

Cancel the common factor of $\cos^{2} (x)$ and $\cos (x)$ .

Tap for more steps...

Step 11.8.1

Factor $\cos (x)$ out of $- \cos^{2} (x)$ .

$\cos (x) + \frac{\cos (x) (- \cos (x))}{\cos (x)} = 0$

Step 11.8.2

Cancel the common factors.

Tap for more steps...

Step 11.8.2.1

Multiply by $1$ .

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

Step 11.8.2.2

Cancel the common factor.

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

Step 11.8.2.3

Rewrite the expression.

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

Step 11.8.2.4

Divide $- \cos (x)$ by $1$ .

$\cos (x) - \cos (x) = 0$

Step 11.9

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

$0 = 0$

Step 12

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 13

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$