Trigonometry Examples

$\cot (x) \csc (x) (1 - \cos (x))$

Step 1

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

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

Step 2

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

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

Step 3

Multiply $\frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\sin (x)}$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 4

Apply the distributive property.

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

Step 5

Simplify the expression.

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Step 5.1

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

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

Step 5.2

Rewrite using the commutative property of multiplication.

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

Step 6

Multiply $- \frac{\cos (x)}{\sin^{2} (x)} \cos (x)$ .

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Add $1$ and $1$ .

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

Step 7

Simplify each term.

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Step 7.1

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

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

Step 7.2

Separate fractions.

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

Step 7.3

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Step 7.4

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

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

Step 7.5

Convert from $\frac{\cos^{2} (x)}{\sin^{2} (x)}$ to $\cot^{2} (x)$ .

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