Trigonometry Examples

$\cot (x) = \csc (2 x) + \cot (2 x)$

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

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

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

Step 2

Simplify the right side.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Rewrite $\csc (2 x)$ in terms of sines and cosines.

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

Step 2.1.2

Rewrite $\cot (2 x)$ in terms of sines and cosines.

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

Step 3

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

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

Step 4

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

Tap for more steps...

Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Apply the distributive property.

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

Step 6

Combine $\sin (x)$ and $\frac{1}{\sin (2 x)}$ .

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

Step 7

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

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

Step 8

Simplify each term.

Tap for more steps...

Step 8.1

Apply the sine double-angle identity.

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

Step 8.2

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

Tap for more steps...

Step 8.2.1

Cancel the common factor.

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

Step 8.2.2

Rewrite the expression.

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

Step 8.3

Use the double-angle identity to transform $\cos (2 x)$ to $2 \cos^{2} (x) - 1$ .

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

Step 8.4

Apply the sine double-angle identity.

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

Step 8.5

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

Tap for more steps...

Step 8.5.1

Cancel the common factor.

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

Step 8.5.2

Rewrite the expression.

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

Step 8.6

Apply the cosine double-angle identity.

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

Step 9

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

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

Step 10

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 10.1

Subtract $\frac{1}{2 \cos (x)}$ from both sides of the equation.

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

Step 10.2

Subtract $\frac{1 - 2 \sin^{2} (x)}{2 \cos (x)}$ from both sides of the equation.

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

Step 11

Simplify the left side.

Tap for more steps...

Step 11.1

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

Tap for more steps...

Step 11.1.1

Simplify terms.

Tap for more steps...

Step 11.1.1.1

Combine the numerators over the common denominator.

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

Step 11.1.1.2

Simplify each term.

Tap for more steps...

Step 11.1.1.2.1

Apply the distributive property.

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

Step 11.1.1.2.2

Multiply $- 1$ by $1$ .

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

Step 11.1.1.2.3

Multiply $- 2$ by $- 1$ .

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

Step 11.1.1.3

Simplify with factoring out.

Tap for more steps...

Step 11.1.1.3.1

Subtract $1$ from $- 1$ .

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

Step 11.1.1.3.2

Factor $- 2$ out of $- 2$ .

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

Step 11.1.1.3.3

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

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

Step 11.1.1.3.4

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

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

Step 11.1.2

Apply pythagorean identity.

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

Step 11.1.3

Simplify terms.

Tap for more steps...

Step 11.1.3.1

Simplify each term.

Tap for more steps...

Step 11.1.3.1.1

Cancel the common factor of $- 2$ and $2$ .

Tap for more steps...

Step 11.1.3.1.1.1

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

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

Step 11.1.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 11.1.3.1.1.2.1

Factor $2$ out of $2 \cos (x)$ .

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

Step 11.1.3.1.1.2.2

Cancel the common factor.

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

Step 11.1.3.1.1.2.3

Rewrite the expression.

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

Step 11.1.3.1.2

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

Tap for more steps...

Step 11.1.3.1.2.1

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

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

Step 11.1.3.1.2.2

Cancel the common factors.

Tap for more steps...

Step 11.1.3.1.2.2.1

Multiply by $1$ .

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

Step 11.1.3.1.2.2.2

Cancel the common factor.

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

Step 11.1.3.1.2.2.3

Rewrite the expression.

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

Step 11.1.3.1.2.2.4

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

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

Step 11.1.3.2

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:

$(- \infty, \infty)$