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

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

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expand (-e^tcos(t)-sin(t))^2

expand

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

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

Solution steps

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

Apply Perfect Square Formula:

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

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

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

Simplify

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

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

Use the Double Angle identity:

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

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

\frac{1}{( x + 1 ) ( y + 1 )}

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

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

sin (3 x) sin (3 (a - x))

[(- 4 + 9) - (+ 21)] + [(+ 8 - 25) + (- 3 - 15)]

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