Simplify sin^2(x-30)+sin^2(x+30)-sin^2x

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thilina-g | College Teacher | (Level 1) Educator

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To simplify this I will use the following.

`(sin(x-30)+sin(x+30))^2 = sin^2(x-30)+sin^2(x+30)+2sin(x-30)sin(x+30)`

Therefore,

`sin^2(x-30)+sin^2(x+30) = (sin(x-30)+sin(x+30))^2 - 2sin(x-30)sin(x+30)`

Therefore,

`sin^2(x-30)+sin^2(x+30) - sin^2(x) =`

`= (sin(x-30)+sin(x+30))^2 - 2sin(x-30)sin(x+30) - sin^2(x)`

`= (2sin(x)cos(-30))^2 - 2sin(x-30)sin(x+30) - sin^2(x)`

`= (2sin(x)cos(30))^2 - 2sin(x-30)sin(x+30) - sin^2(x)`

`= 4sin^2(x)cos^2(30) - 2sin(x-30)sin(x+30) - sin^2(x)`

`= 4sin^2(x)cos^2(30) -(cos(-60)-cos(2x))- sin^2(x)`

`= 4sin^2(x)cos^2(30)-cos(60)+cos(2x)- sin^2(x)`

`= 4sin^2(x)cos^2(30)-cos(60)+1-2sin^2(x)- sin^2(x)`

`= 4sin^2(x)xx3/4-cos(60)+1-2sin^2(x)- sin^2(x)`

`= 3sin^2(x)-cos(60)+1-2sin^2(x)- sin^2(x)`

`= 1-1/2`

`= 1/2`

 

Therefore,

`sin^2(x-30)+sin^2(x+30) - sin^2(x) = 1/2`

 

 

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