Trigonometry Examples

$\frac{\sin (x)}{\cos (x)} + \frac{\cos (x)}{\sin (x)} = \csc (x) \cdot \sec (x)$

Step 1

Subtract $\csc (x) \cdot \sec (x)$ from both sides of the equation.

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

Step 2

Simplify $\frac{\sin (x)}{\cos (x)} + \frac{\cos (x)}{\sin (x)} - \csc (x) \cdot \sec (x)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

Multiply $\frac{1}{\cos (x)}$ by $\frac{1}{\sin (x)}$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Separate fractions.

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

Step 2.2.4

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

$\tan (x) + \cot (x) - (\frac{1}{\sin (x)} \sec (x)) = 0$

Step 2.2.5

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\tan (x) + \cot (x) - \csc (x) \sec (x) = 0$

Step 3

Set the argument in $\tan (x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$x = \frac{π}{2} + π n$ , for any integer $n$

Step 4

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 5

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 $0$ .

${x | x = \frac{π}{2} + π n, x = π n}$ $n$ , for any integer $n$

Step 6