Trigonometry Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Apply pythagorean identity.

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

Step 2.8

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

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Step 2.8.1

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

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

Step 2.8.2

Cancel the common factors.

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Step 2.8.2.1

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

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

Step 2.8.2.2

Cancel the common factor.

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

Step 2.8.2.3

Rewrite the expression.

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

Step 2.9

Separate fractions.

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

Step 2.10

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

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

Step 2.11

Divide $- 1$ by $1$ .

$- \cot (x) = 0$

Step 3

Set the argument in $\cot (x)$ equal to $π n$ to find where the expression is undefined.

$x = π n$ , for any integer $n$

Step 4

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

${x | x = π n}$ $n$ , for any integer $n$

Step 5