Trigonometry Examples

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

Step 1

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

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

Step 2

Since $\csc (- x)$ is an odd function, rewrite $\csc (- x)$ as $- \csc (x)$ .

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

Step 3

Rewrite using the commutative property of multiplication.

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Add $1$ and $1$ .

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

Step 6

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Add $1$ and $1$ .

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

Step 8

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

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