Simplify (1-(sin(x)-cos(x))^2)/(2cos(x))

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

Step 1

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

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

Step 2

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 $b = \sin (x) - \cos (x)$ .

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

Step 3

Simplify.

Apply the distributive property.

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

Multiply $- - \cos (x)$ .

Multiply $- 1$ by $- 1$ .

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

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

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

Simplify (1-(sin(x)-cos(x))^2)/(2cos(x))

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