Trigonometry Examples

$\frac{\cot (x) - \cot (x) \sec^{2} (x)}{\tan (x) \cot (x)}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $\cot (x)$ out of $\cot (x) - \cot (x) \sec^{2} (x)$ .

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Step 1.1.1

Multiply by $1$ .

$\frac{\cot (x) \cdot 1 - \cot (x) \sec^{2} (x)}{\tan (x) \cot (x)}$

Step 1.1.2

Factor $\cot (x)$ out of $- \cot (x) \sec^{2} (x)$ .

$\frac{\cot (x) \cdot 1 + \cot (x) (- 1 \sec^{2} (x))}{\tan (x) \cot (x)}$

Step 1.1.3

Factor $\cot (x)$ out of $\cot (x) \cdot 1 + \cot (x) (- 1 \sec^{2} (x))$ .

$\frac{\cot (x) (1 - 1 \sec^{2} (x))}{\tan (x) \cot (x)}$

Step 1.2

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

$\frac{\cot (x) (1^{2} - 1 \sec^{2} (x))}{\tan (x) \cot (x)}$

Step 1.3

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 $b = \sec (x)$ .

$\frac{\cot (x) (1 + \sec (x)) (1 - \sec (x))}{\tan (x) \cot (x)}$

Step 2

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

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Step 2.1

Cancel the common factor.

$\frac{\cot (x) (1 + \sec (x)) (1 - \sec (x))}{\tan (x) \cot (x)}$

Step 2.2

Rewrite the expression.

$\frac{(1 + \sec (x)) (1 - \sec (x))}{\tan (x)}$