Trigonometry Examples

$\frac{\cot^{2} (x)}{\csc (x)} = \csc (x) - \sin (x)$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

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

$\frac{\cot^{2} (x)}{\csc (x)} - \csc (x) = - \sin (x)$

Step 1.2

Add $\sin (x)$ to both sides of the equation.

$\frac{\cot^{2} (x)}{\csc (x)} - \csc (x) + \sin (x) = 0$

Step 2

Simplify $\frac{\cot^{2} (x)}{\csc (x)} - \csc (x) + \sin (x)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Separate fractions.

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

Step 2.1.2

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

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

Step 2.1.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

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

Step 2.1.4

Multiply $\sin (x)$ by $1$ .

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

Step 2.1.5

Divide $\cot^{2} (x)$ by $1$ .

$\cot^{2} (x) \sin (x) - \csc (x) + \sin (x) = 0$

Step 2.1.6

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

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

Step 2.1.7

Apply the product rule to $\frac{\cos (x)}{\sin (x)}$ .

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

Step 2.1.8

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

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Step 2.1.8.1

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

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

Step 2.1.8.2

Cancel the common factor.

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

Step 2.1.8.3

Rewrite the expression.

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

Step 2.1.9

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

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Apply pythagorean identity.

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

Step 2.9

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

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Step 2.9.1

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

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

Step 2.9.2

Cancel the common factors.

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Step 2.9.2.1

Multiply by $1$ .

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

Step 2.9.2.2

Cancel the common factor.

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

Step 2.9.2.3

Rewrite the expression.

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

Step 2.9.2.4

Divide $- \sin (x)$ by $1$ .

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

Step 2.10

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

$0 = 0$

Step 3

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.