Trigonometry Examples

$\ln (\cot (x)) = - \ln (\tan (x))$

Step 1

Simplify $- \ln (\tan (x))$ by moving $- 1$ inside the logarithm.

$\ln (\cot (x)) = \ln (\tan^{- 1} (x))$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

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

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Simplify the right side.

Tap for more steps...

Step 3.1.1

Simplify $\tan^{- 1} (x)$ .

Tap for more steps...

Step 3.1.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.1.1.2

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

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

Step 3.1.1.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 3.1.1.4

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\cot (x) = \cot (x)$

Step 3.2

For the two functions to be equal, the arguments of each must be equal.

$x = x$

Step 3.3

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

Tap for more steps...

Step 3.3.1

Subtract $x$ from both sides of the equation.

$x - x = 0$

Step 3.3.2

Subtract $x$ from $x$ .

$0 = 0$

Step 3.4

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

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$