the number of all five tuples (a1,a2,a3,a4,a5) such that a1+a2sinx+a3cosx+a4sin2x+a5cos2x=0 holds for x belongs to R is
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Note that a six + b cos x = r sin (x+theta) where
r = sqrt(a*a+b*b) and theta = arctan(b/a)
So
a1+a2sinx+a3cosx+a4sin2x+a5cos2x
= a1 + A sin (x+theta1) + B sin (2x + theta2)
If it is zero for all values of x, then a1 = A = B = 0
A = B = 0 => a2 = a3 = a4 = a5 = 0
r = sqrt(a*a+b*b) and theta = arctan(b/a)
So
a1+a2sinx+a3cosx+a4sin2x+a5cos2x
= a1 + A sin (x+theta1) + B sin (2x + theta2)
If it is zero for all values of x, then a1 = A = B = 0
A = B = 0 => a2 = a3 = a4 = a5 = 0
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