What is expand (-e^{-x}sin(x)+e^{-x}cos(x))^2 ?

The solution to expand (-e^{-x}sin(x)+e^{-x}cos(x))^2 is e^{-2x}(-sin(2x)+1)

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expand (-e^{-x}sin(x)+e^{-x}cos(x))^2

expand

(- e^{- x} sin (x) + e^{- x} cos (x))^{2}

e^{- 2 x} (- sin (2 x) + 1)

Solution steps

(- e^{- x} sin (x) + e^{- x} cos (x))^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = - e^{- x} sin (x), b = e^{- x} cos (x)

= (- e^{- x} sin (x))^{2} + 2 (- e^{- x} sin (x)) e^{- x} cos (x) + (e^{- x} cos (x))^{2}

Simplify

(- e^{- x} sin (x))^{2} + 2 (- e^{- x} sin (x)) e^{- x} cos (x) + (e^{- x} cos (x))^{2} : e^{- 2 x} sin^{2} (x) - 2 e^{- 2 x} sin (x) cos (x) + e^{- 2 x} cos^{2} (x)

= e^{- 2 x} sin^{2} (x) - 2 e^{- 2 x} sin (x) cos (x) + e^{- 2 x} cos^{2} (x)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

= cos^{2} (x) e^{- 2 x} + sin^{2} (x) e^{- 2 x} - sin (2 x) e^{- 2 x}

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

= e^{- 2 x} (- sin (2 x) + 1)

\frac{12}{x ( x - 4 )}

(2, - 3) y (4, 2)

(2 x^{3} - x y^{2} - 2 y + 3) d x + (x^{2} y + 2 x)

(a + \frac{1}{4}) (a - \frac{1}{4})

(3 + 5 x)^{2}

What is expand (-e^{-x}sin(x)+e^{-x}cos(x))^2 ?
The solution to expand (-e^{-x}sin(x)+e^{-x}cos(x))^2 is e^{-2x}(-sin(2x)+1)