Algebra Examples

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

Step 1

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

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

Step 2

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

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

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 $b = \sin (θ) - \cos (θ)$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $- - \cos (θ)$ .

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Step 4.2.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2.2

Multiply $\cos (θ)$ by $1$ .

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