What is expand (sin((pix)/L)sin((2piy)/L)-sin((2pix)/L)sin((piy)/L))^2 ?

The solution to expand (sin((pix)/L)sin((2piy)/L)-sin((2pix)/L)sin((piy)/L))^2 is

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expand (sin((pix)/L)sin((2piy)/L)-sin((2pix)/L)sin((piy)/L))^2

expand

(sin (\frac{π x}{L}) sin (\frac{2 π y}{L}) - sin (\frac{2 π x}{L}) sin (\frac{π y}{L}))^{2}

sin^{2} (\frac{π x}{L}) sin^{2} (\frac{2 π y}{L}) - 2 sin (\frac{π x}{L}) sin (\frac{π y}{L}) sin (\frac{2 π y}{L}) sin (\frac{2 π x}{L}) + sin^{2} (\frac{π y}{L}) sin^{2} (\frac{2 π x}{L})

Solution steps

(sin (\frac{π x}{L}) sin (\frac{2 π y}{L}) - sin (\frac{2 π x}{L}) sin (\frac{π y}{L}))^{2}

= (sin (\frac{π x}{L}) sin (\frac{2 π y}{L}) - sin (\frac{π y}{L}) sin (\frac{2 π x}{L}))^{2}

Apply Perfect Square Formula:

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

a = sin (\frac{π x}{L}) sin (\frac{2 π y}{L}), b = sin (\frac{2 π x}{L}) sin (\frac{π y}{L})

= (sin (\frac{π x}{L}) sin (\frac{2 π y}{L}))^{2} - 2 sin (\frac{π x}{L}) sin (\frac{2 π y}{L}) sin (\frac{2 π x}{L}) sin (\frac{π y}{L}) + (sin (\frac{2 π x}{L}) sin (\frac{π y}{L}))^{2}

Simplify

(sin (\frac{π x}{L}) sin (\frac{2 π y}{L}))^{2} - 2 sin (\frac{π x}{L}) sin (\frac{2 π y}{L}) sin (\frac{2 π x}{L}) sin (\frac{π y}{L}) + (sin (\frac{2 π x}{L}) sin (\frac{π y}{L}))^{2} : sin^{2} (\frac{π x}{L}) sin^{2} (\frac{2 π y}{L}) - 2 sin (\frac{π x}{L}) sin (\frac{π y}{L}) sin (\frac{2 π y}{L}) sin (\frac{2 π x}{L}) + sin^{2} (\frac{π y}{L}) sin^{2} (\frac{2 π x}{L})

= sin^{2} (\frac{π x}{L}) sin^{2} (\frac{2 π y}{L}) - 2 sin (\frac{π x}{L}) sin (\frac{π y}{L}) sin (\frac{2 π y}{L}) sin (\frac{2 π x}{L}) + sin^{2} (\frac{π y}{L}) sin^{2} (\frac{2 π x}{L})

\frac{1 - 3 e ^{- s} + s ^{2} + 9 s}{s ( s + 4 ) ^{2}}

(32 x - 3)^{2}

p u n t os (- 3, 2) y (6, - 4)

(2 + 7 i)^{6}

What is expand (sin((pix)/L)sin((2piy)/L)-sin((2pix)/L)sin((piy)/L))^2 ?
The solution to expand (sin((pix)/L)sin((2piy)/L)-sin((2pix)/L)sin((piy)/L))^2 is