What is expand (e^t*sin(3t)+e^t*cos(3t))^2 ?

The solution to expand (e^t*sin(3t)+e^t*cos(3t))^2 is e^{2t}(sin(6t)+1)

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expand (e^tsin(3t)+e^tcos(3t))^2

expand

(e^{t} \cdot sin (3 t) + e^{t} \cdot cos (3 t))^{2}

e^{2 t} (sin (6 t) + 1)

Solution steps

(e^{t} sin (3 t) + e^{t} cos (3 t))^{2}

Apply Perfect Square Formula:

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

a = e^{t} sin (3 t), b = e^{t} cos (3 t)

= (e^{t} sin (3 t))^{2} + 2 e^{t} sin (3 t) e^{t} cos (3 t) + (e^{t} cos (3 t))^{2}

Simplify

(e^{t} sin (3 t))^{2} + 2 e^{t} sin (3 t) e^{t} cos (3 t) + (e^{t} cos (3 t))^{2} : e^{2 t} sin^{2} (3 t) + 2 e^{2 t} sin (3 t) cos (3 t) + e^{2 t} cos^{2} (3 t)

= e^{2 t} sin^{2} (3 t) + 2 e^{2 t} sin (3 t) cos (3 t) + e^{2 t} cos^{2} (3 t)

Use the Double Angle identity:

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

= cos^{2} (3 t) e^{2 t} + sin^{2} (3 t) e^{2 t} + sin (2 \cdot 3 t) e^{2 t}

Use the Pythagorean identity:

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

= e^{2 t} (sin (6 t) + 1)

Factor

cos^{2} (3 t) e^{2 t} + sin^{2} (3 t) e^{2 t} + sin (2 \cdot 3 t) e^{2 t} : e^{2 t} (cos^{2} (3 t) + sin^{2} (3 t) + sin (6 t))

= e^{2 t} (sin (6 t) + 1)

\frac{4 ( 9 ^{20} - 8 ^{20} )}{3 ^{38}}

\frac{( x - 1 ) ^{2}}{( x + 1 )}

x^{2} (e^{3 x^{2}} - e^{x})

\frac{1}{2} (e^{4 t} + e^{- 2 t}) (6 t + 1)

\frac{1}{( 2 s + 1 ) ^{2} ( s - j w )}

What is expand (e^t*sin(3t)+e^t*cos(3t))^2 ?
The solution to expand (e^t*sin(3t)+e^t*cos(3t))^2 is e^{2t}(sin(6t)+1)