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

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

Step 1

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

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

Step 2

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

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

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 $b = \cos (θ) - \sin (θ)$ .

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

Step 4

Simplify.

Apply the distributive property.

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

Multiply $- - \sin (θ)$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $\sin (θ)$ by $1$ .

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

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

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