Trigonometry Examples

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

Step 1

Subtract $\frac{\cos^{2} (a)}{\sin (a)}$ from both sides of the equation.

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

Step 2

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

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

Since $\sin (- a)$ is an odd function, rewrite $\sin (- a)$ as $- \sin (a)$ .

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

Apply pythagorean identity.

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Multiply by $1$ .

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

Step 2.4.2.2

Cancel the common factor.

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

Step 2.4.2.3

Rewrite the expression.

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

Step 2.4.2.4

Divide $\sin (a)$ by $1$ .

$- \sin (a) + \sin (a) = 0$

Step 2.5

Add $- \sin (a)$ and $\sin (a)$ .

$0 = 0$

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.