Trigonometry Examples

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

Step 1

Set the numerator equal to zero.

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

Step 2

Solve the equation for $x$ .

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Step 2.1

Divide each term in the equation by $\cos (x)$ .

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

Step 2.2

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

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Separate fractions.

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

Step 2.6

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

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

Step 2.7

Divide $- 1$ by $1$ .

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

Step 2.8

Separate fractions.

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

Step 2.9

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 2.10

Divide $0$ by $1$ .

$\tan (x) - \tan (x) = 0 \sec (x)$

Step 2.11

Multiply $0$ by $\sec (x)$ .

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

Step 2.12

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

$0 = 0$

Step 2.13

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

All real numbers

Step 3

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$