Trigonometry Examples

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

Step 1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.1

Reorder $\sec (x)$ and $\cot (x)$ .

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

Step 1.2

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

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

Step 1.3

Cancel the common factors.

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

Step 2

To write $- \sin (x)$ as a fraction with a common denominator, multiply by $\frac{\sin (x)}{\sin (x)}$ .

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

Step 3

Simplify terms.

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Step 3.1

Combine $- \sin (x)$ and $\frac{\sin (x)}{\sin (x)}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $1$ as $1^{2}$ .

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

Step 4.2

Rewrite $\sin (x) \sin (x)$ as $\sin^{2} (x)$ .

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

Step 4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \sin (x)$ .

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