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

The solution to expand (-e^{2t}sin(t)+e^{2t}cos(t))^2 is e^{4t}(-sin(2t)+1)

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

expand

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

e^{4 t} (- sin (2 t) + 1)

Solution steps

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

Apply Perfect Square Formula:

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

a = - e^{2 t} sin (t), b = e^{2 t} cos (t)

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

Simplify

(- e^{2 t} sin (t))^{2} + 2 (- e^{2 t} sin (t)) e^{2 t} cos (t) + (e^{2 t} cos (t))^{2} : e^{4 t} sin^{2} (t) - 2 e^{4 t} sin (t) cos (t) + e^{4 t} cos^{2} (t)

= e^{4 t} sin^{2} (t) - 2 e^{4 t} sin (t) cos (t) + e^{4 t} cos^{2} (t)

Use the Double Angle identity:

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

= cos^{2} (t) e^{4 t} + sin^{2} (t) e^{4 t} - sin (2 t) e^{4 t}

Use the Pythagorean identity:

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

= e^{4 t} (- sin (2 t) + 1)

\frac{3 ( x ^{2} - 9 x + 20 )}{2}

(x^{2} - 5 e x + 3) (x^{3} + 2 x + 1)

1 + (\frac{( x ^{4} + 9 )}{( 18 x )})^{2}

(bi + c j + d k)^{2}

(1 + e^{x}) d x

What is expand (-e^{2t}sin(t)+e^{2t}cos(t))^2 ?
The solution to expand (-e^{2t}sin(t)+e^{2t}cos(t))^2 is e^{4t}(-sin(2t)+1)