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, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \cos (x) + \sin (x)$ .

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

Step 4

Apply the distributive property.

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