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sin^2 x +sinx = 0
To find the solution for x
We try writing the leftside in factors.
(sinx)^2+sinx = 0
sinx(sinx+1) = 0 . Or
Sofirst factor, sinx = 0 or Or second factor sinx +1 = 0.
By Sinx = 0 , x = n*pi where n belongs tothe set ofintegers numbers 0,1,2,3................................(a)
By second factor, sinx +1 = 0 Or sinx = -1. Or x= (2n+1)pi, where n is 0,1,2,3,4,..........................(b)
Note: (sinx)^2+sinx = 0 is not an identity. It is an equation true for some particular x. An identy is an equation which holds true for all x. If you test for any x other than the solution set for x at (a) or at (b) , the equation does not hold good. Example: Whenx=pi/2 radians or 90 degrees, then sin (pi/2)=1. So, [sin(pi/2)]^2 +sin(pi/2) = 1+1 =2 is the value of left side of the given equation.But the right side is zero. So the equation does not hold good. Therefore this is only an equation and not an identity.
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