Precalculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

Combine $\cos^{2} (x)$ and $\frac{1}{\sin (x)}$ .

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

Step 2.1.3

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

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

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$