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if x and y are the solutions of 12 sinx + 5 cos x= 2 y^2-8y+21, the value of 3864 cot(xy/2) is?
LHS=12 sinx+5cosx</ sqrt(144+25)=12sinx+5cosx</13LHS</13RHS=2(y^2-4y)+21=2(y^2-4y+4)-8+21=2(y-2)^2+13hence RHS>/13Roots of the equation exist if LHS=RHS=13=> 2y^2-8y+21=13=>2y^2-8y+8=0=>y^2-4y+4=0=>(y-2)^2=0=>y=212sinx+5cosx=13(12/13)sinx+(5/13)cosx=1cos@sinx+sin@cosx=1 where @=arctan(5/12)sin(x+@)=1=sin(pi/2)x+arctan(5/12)=pi/2x=pi/2-arctan(5/12)=arccot(5/12)cotx=5/12Hence 3864 cot(xy/2)=3864 cotx=3864(5/12)=1610Thanks & RegardsRinkoo GuptaAskIITians Faculty
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