For any `a,b,c,x,y >0` prove that : `2/3tan^(- 1)((3a b^2-a^3)/(b^3-3a^2b))+2/3tan^(- 1)((3x y^2-x^3)/(y^3-3x^2y))=tan^(- 1)\ (2alphabeta)/(a^2-beta^2)` where `alpha=-a x+b y ,beta=b x+a y`

For any a ,\ b ,\ x ,\ y >0 , prove that : 2/3tan^(-1)((3a b^2-a^3)/(b^3-3a^2b))+2/3tan^(-1)((3x y^2-x^3)/(y^3-3x^2y))=tan^(-1)(2alphabeta)/(alpha^2-beta^2) , where alpha=-a x+b y beta=b x+a ydot

For any a,b,c,x,y>0 prove that :(2)/(3)tan^(-1)((3ab^(2)-a^(3))/(b^(3)-3a^(2)b))+(2)/(3)tan^(-1)((3xy^(2)-x^(3))/(y^(3)-3x^(2)y))=tan^(-1)(2 alpha beta)/(a^(2)-beta^(2)) where alpha=-ax+by,beta=bx+ay

If alpha=tan^(-1)((sqrt(3)x)/(2y-x)),beta=tan^(-1)((2x-y)/(sqrt(3y)))" then "alpha-beta=

If alpha = tan^(-1) ((xsqrt(3))/(2y - x)) and beta = tan^(-1) ((2x - y)/(ysqrt(3))) evaluate : alpha - beta .

If alpha=tan^(-1)((sqrt(3)x)/(2y-x)) , beta=tan^(-1)((2x-y)/(sqrt(3)y)) , then alpha-beta= pi/6 (b) pi/3 (c) pi/2 (d) -pi/3

If alpha=tan^(-1)((sqrt(3)x)/(2y-x)) , beta=tan^(-1)((2x-y)/(sqrt(3)y)) , then alpha-beta= (a) pi/6 (b) pi/3 (c) pi/2 (d) -pi/3

If x : a = y : b = z : c , then prove that x^(3)/a^(2) + y^(3)/b^(2) + z^(3)/c^(2) = ((x+y+z)^(3))/((a+b+c)^(2))

If ((x^(-1)y^(2))/(x^(3)y^(-2)))^(1)+((x^(6)y^(-3))/(x^(2)y^(3)))^((1)/(2))=x^(a)y^(b) prove that a+b=-1, where x and y are different positive primes.

If ((x^(-1)y^2)/(x^3y^(-2)))^1.\ ((x^6\ y^(-3))/(x^2\ y^3))^(1/2)=x^a y^b , prove that a+b=-1 , where x\ a n d\ y are different positive primes.