Simplify (1-(sin(theta)+cos(theta))(sin(theta)+cos(theta)))/(sin(theta)cos(theta))

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

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

Apply the distributive property.

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

Simplify (1-(sin(theta)+cos(theta))(sin(theta)+cos(theta)))/(sin(theta)cos(theta))

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