Trigonometry Examples

$\cot (x) (\sin (x) - \sec (x))$

Step 1

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

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Cancel the common factor of $\sin (x)$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 4

Simplify each term.

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Step 4.1

Cancel the common factor of $\cos (x)$ .

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Step 4.1.1

Move the leading negative in $- \frac{\cos (x)}{\sin (x)}$ into the numerator.

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

Step 4.1.2

Factor $\cos (x)$ out of $- \cos (x)$ .

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

Step 4.1.3

Cancel the common factor.

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

Step 4.1.4

Rewrite the expression.

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

Step 4.2

Move the negative in front of the fraction.

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

Step 5

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

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