Trigonometry Examples

$\frac{\csc (x)}{\tan (x) \cot (x)} = \cos (x)$

Step 1

Subtract $\cos (x)$ from both sides of the equation.

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

Step 2

Simplify $\frac{\csc (x)}{\tan (x) \cot (x)} - \cos (x)$ .

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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.

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

Step 2.1.2

Simplify the denominator.

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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

Step 2.1.3

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

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

Step 2.1.4

Simplify the denominator.

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Step 2.1.4.1

Reduce the expression $\frac{\sin (x) \cos (x)}{\cos (x) \sin (x)}$ by cancelling the common factors.

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Step 2.1.4.1.1

Cancel the common factor.

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

Step 2.1.4.1.2

Rewrite the expression.

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

Step 2.1.4.2

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

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Step 2.1.4.2.1

Cancel the common factor.

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

Step 2.1.4.2.2

Rewrite the expression.

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

Step 2.1.5

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

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

Step 2.2

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

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

Step 3

Set the argument in $\csc (x)$ 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