Prove that : `tan^(-1), (a^3 -b^3)/(1+a^3 b^3) + tan^(-1), (b^3 - c^3)/(1+b^3 c^3) + tan^(-1), (c^3 - a^3)/(1+c^3 a^3) = 0`

Prove that : tan^(-1)( (2a-b)/(bsqrt(3))) +tan^(-1)((2b-a)/(asqrt(3))) = pi/3

Prove that : |{:(1,b,c),(b,c,a),(c,a,b):}|=3 a b c-a^(3)-b^(3)-c^(3)

Prove that |{:(1, 1, 1),(a, b, c),(a^(3), b^(3), c^(3)):}|=(a-b)(b-c)(c-a)(a+b+c)

If the points ((a^(3))/(a-1),(a^(2)-3)/(a-1)),((b^(3))/(b-1),(b^(3)-3)/(b-1)) and ((c^(3))/(c-1),(c^(3)-3)/(c-1)) where a,b,c are different from 1 lie on the line lx+my+n=0a+b+c=(m)/(l)ab+bc+ca+(n)/(l)=0abc=((3m+n))/(l)abc-(bc+ca+ab)+3(a+b+c)=0

Prove that (a^(8)+b^(8)+c^(8))/(a^(3)b^(3)c^(3))>(1)/(a)+(1)/(b)+(1)/(c)

If the points ((a^(3))/(a-1),(a^(2)-3)/(a-1))((b^(3))/(b-1),(b^(2)-3)/(b-1)),((c^(3))/(c-1),(c^(2)-3)/(c-1)) are a!=1,b!=1,c!=1, then find the value of abc-(ab+bc+ca)+3(a+b+c)

Prove that: (i) |{:(,1,1,1),(,a,b,c),(,a^(3),b^(3),c^(3)):}|=(a-b)(b-c)(c-a)(a+b+c)