Trigonometry Examples

$\frac{1}{\sin^{3} (x)} - \frac{\cot^{2} (x)}{\sin (x)}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Simplify the numerator.

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Step 1.1.1.1

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

$\frac{1}{\sin^{3} (x)} - \frac{{(\frac{\cos (x)}{\sin (x)})}^{2}}{\sin (x)}$

Step 1.1.1.2

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

$\frac{1}{\sin^{3} (x)} - \frac{\frac{\cos^{2} (x)}{\sin^{2} (x)}}{\sin (x)}$

Step 1.1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{\sin^{3} (x)} - (\frac{\cos^{2} (x)}{\sin^{2} (x)} \cdot \frac{1}{\sin (x)})$

Step 1.1.3

Combine.

$\frac{1}{\sin^{3} (x)} - \frac{\cos^{2} (x) \cdot 1}{\sin^{2} (x) \sin (x)}$

Step 1.1.4

Multiply $\sin^{2} (x)$ by $\sin (x)$ by adding the exponents.

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Step 1.1.4.1

Multiply $\sin^{2} (x)$ by $\sin (x)$ .

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Step 1.1.4.1.1

Raise $\sin (x)$ to the power of $1$ .

$\frac{1}{\sin^{3} (x)} - \frac{\cos^{2} (x) \cdot 1}{\sin^{2} (x) \sin^{1} (x)}$

Step 1.1.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{\sin^{3} (x)} - \frac{\cos^{2} (x) \cdot 1}{{\sin (x)}^{2 + 1}}$

Step 1.1.4.2

Add $2$ and $1$ .

$\frac{1}{\sin^{3} (x)} - \frac{\cos^{2} (x) \cdot 1}{\sin^{3} (x)}$

Step 1.1.5

Multiply $\cos^{2} (x)$ by $1$ .

$\frac{1}{\sin^{3} (x)} - \frac{\cos^{2} (x)}{\sin^{3} (x)}$

Step 1.2

Combine the numerators over the common denominator.

$\frac{1 - \cos^{2} (x)}{\sin^{3} (x)}$

Step 2

Apply pythagorean identity.

$\frac{\sin^{2} (x)}{\sin^{3} (x)}$

Step 3

Cancel the common factor of $\sin^{2} (x)$ and $\sin^{3} (x)$ .

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Step 3.1

Multiply by $1$ .

$\frac{\sin^{2} (x) \cdot 1}{\sin^{3} (x)}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $\sin^{2} (x)$ out of $\sin^{3} (x)$ .

$\frac{\sin^{2} (x) \cdot 1}{\sin^{2} (x) \sin (x)}$

Step 3.2.2

Cancel the common factor.

$\frac{\sin^{2} (x) \cdot 1}{\sin^{2} (x) \sin (x)}$

Step 3.2.3

Rewrite the expression.

$\frac{1}{\sin (x)}$

Step 4

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

$\csc (x)$