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)

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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)