What is expand (4e^xcos(x)-4e^xsin(x))^2 ?

The solution to expand (4e^xcos(x)-4e^xsin(x))^2 is 16e^{2x}(1-sin(2x))

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expand (4e^xcos(x)-4e^xsin(x))^2

expand

(4 e^{x} cos (x) - 4 e^{x} sin (x))^{2}

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

Solution steps

(4 e^{x} cos (x) - 4 e^{x} sin (x))^{2}

Apply Perfect Square Formula:

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

a = 4 e^{x} cos (x), b = 4 e^{x} sin (x)

= (4 e^{x} cos (x))^{2} - 2 \cdot 4 e^{x} cos (x) \cdot 4 e^{x} sin (x) + (4 e^{x} sin (x))^{2}

Simplify

(4 e^{x} cos (x))^{2} - 2 \cdot 4 e^{x} cos (x) \cdot 4 e^{x} sin (x) + (4 e^{x} sin (x))^{2} : 16 e^{2 x} cos^{2} (x) - 32 e^{2 x} cos (x) sin (x) + 16 e^{2 x} sin^{2} (x)

= 16 e^{2 x} cos^{2} (x) - 32 e^{2 x} cos (x) sin (x) + 16 e^{2 x} sin^{2} (x)

Use the Pythagorean identity:

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

= 16 e^{2 x} (- 2 cos (x) sin (x) + 1)

Use the Double Angle identity:

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

= 16 e^{2 x} (1 - sin (2 x))

Factor

16 e^{2 x} cos^{2} (x) - 32 e^{2 x} cos (x) sin (x) + 16 e^{2 x} sin^{2} (x) : 16 e^{2 x} (cos^{2} (x) - 2 cos (x) sin (x) + sin^{2} (x))

= 16 e^{2 x} (- sin (2 x) + 1)

- 10 (- 3)

\frac{4}{x ( x + 1 ) ( x + 2 )}

\frac{( x ^{2} + 4 x + 1 )}{( x - 1 ) ( x + 1 ) ( x + 3 )}

\frac{u ( t - 1 )}{e ^{t a}}

Δ^{3} y

What is expand (4e^xcos(x)-4e^xsin(x))^2 ?
The solution to expand (4e^xcos(x)-4e^xsin(x))^2 is 16e^{2x}(1-sin(2x))