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

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

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expand (e^tsin(t)-cos(t))^2

expand

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

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

Solution steps

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

Apply Perfect Square Formula:

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

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

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

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

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

Use the Double Angle identity:

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

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

\frac{d}{d x} ((f (x, y)) (x^{2} + y^{2} - 1))

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

(2 - x) e^{x} - 1

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

3^{(x - 1)}

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