What is expand (7e^tcos(t)+7e^tsin(t))^2 ?

The solution to expand (7e^tcos(t)+7e^tsin(t))^2 is 49e^{2t}(1+sin(2t))

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expand (7e^tcos(t)+7e^tsin(t))^2

expand

(7 e^{t} cos (t) + 7 e^{t} sin (t))^{2}

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

Solution steps

(7 e^{t} cos (t) + 7 e^{t} sin (t))^{2}

Apply Perfect Square Formula:

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

a = 7 e^{t} cos (t), b = 7 e^{t} sin (t)

= (7 e^{t} cos (t))^{2} + 2 \cdot 7 e^{t} cos (t) \cdot 7 e^{t} sin (t) + (7 e^{t} sin (t))^{2}

Simplify

(7 e^{t} cos (t))^{2} + 2 \cdot 7 e^{t} cos (t) \cdot 7 e^{t} sin (t) + (7 e^{t} sin (t))^{2} : 49 e^{2 t} cos^{2} (t) + 98 e^{2 t} cos (t) sin (t) + 49 e^{2 t} sin^{2} (t)

= 49 e^{2 t} cos^{2} (t) + 98 e^{2 t} cos (t) sin (t) + 49 e^{2 t} sin^{2} (t)

Use the Pythagorean identity:

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

= 49 e^{2 t} (2 cos (t) sin (t) + 1)

Use the Double Angle identity:

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

= 49 e^{2 t} (1 + sin (2 t))

Factor

49 e^{2 t} cos^{2} (t) + 98 e^{2 t} cos (t) sin (t) + 49 e^{2 t} sin^{2} (t) : 49 e^{2 t} (cos^{2} (t) + 2 cos (t) sin (t) + sin^{2} (t))

= 49 e^{2 t} (sin (2 t) + 1)

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What is expand (7e^tcos(t)+7e^tsin(t))^2 ?
The solution to expand (7e^tcos(t)+7e^tsin(t))^2 is 49e^{2t}(1+sin(2t))