What is expand (3e^tcos(t)-3e^tsin(t))^2 ?

The solution to expand (3e^tcos(t)-3e^tsin(t))^2 is 9e^{2t}(1-sin(2t))

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

expand

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

9 e^{2 t} (1 - sin (2 t))

Solution steps

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

Apply Perfect Square Formula:

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

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

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

Simplify

(3 e^{t} cos (t))^{2} - 2 \cdot 3 e^{t} cos (t) \cdot 3 e^{t} sin (t) + (3 e^{t} sin (t))^{2} : 9 e^{2 t} cos^{2} (t) - 18 e^{2 t} cos (t) sin (t) + 9 e^{2 t} sin^{2} (t)

= 9 e^{2 t} cos^{2} (t) - 18 e^{2 t} cos (t) sin (t) + 9 e^{2 t} sin^{2} (t)

Use the Pythagorean identity:

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

= 9 e^{2 t} (- 2 cos (t) sin (t) + 1)

Use the Double Angle identity:

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

= 9 e^{2 t} (1 - sin (2 t))

Factor

9 e^{2 t} cos^{2} (t) - 18 e^{2 t} cos (t) sin (t) + 9 e^{2 t} sin^{2} (t) : 9 e^{2 t} (cos^{2} (t) - 2 cos (t) sin (t) + sin^{2} (t))

= 9 e^{2 t} (- sin (2 t) + 1)

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\frac{867 ( 2 + 3 ) 2 - 3}{1960}

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\frac{x ^{3} - 6 x ^{2} + 12 x - 8}{x - 2}

What is expand (3e^tcos(t)-3e^tsin(t))^2 ?
The solution to expand (3e^tcos(t)-3e^tsin(t))^2 is 9e^{2t}(1-sin(2t))