Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $- - \sin (x)$ .

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Step 4.2.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2.2

Multiply $\sin (x)$ by $1$ .

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