If `|a| lt 1, |b| lt 1`, then show that
`a(a+b) + a^(2) (a^(2) + b^(2)) + a^(3)(a^(3) + b^(3)) +… = (a^(2))/(1-a^(2)) + (ab)/(1- ab)`

If a + b = 5, a - b = 1 then let's show that. 8ab (a^2 + b^2) = 624

If |x-a| lt l_(1) and |x-b| lt l_(2) , show that |a-b| lt l_(1) + l_(2) .

((a^(2)b^(-2))/(a^(-2)b^(2)))^(3)xx((ab^(-1))/(a^(-1)b))^(-2)

Factorise a ^(3) - 3a^(2) b + 3ab ^(2) -b ^(3)

If a = (sqrt5 + 1)/(sqrt5 + 1) and b = (sqrt5 -1)/(sqrt5 + 1) , then find the value of (a) (a^(2) + ab + b^(2))/(a^(2) - ab + b^(2)) (b) ((a -b)^(3))/((a + b)^(3)) (c) (3a^(2) + 5ab + b^(2))/(3a^(2) - 5ab + b^(2)) (d) (a^(3) + b^(3))/(a^(3) - b^(3))

If a, b, c are in G.P., show that a^(2)b^(2)c^(2)((1)/(a^(3))+(1)/(b^(3))+(1)/(c^(3))) = a^(3) + b^(3) + c^(3) .

if a lt c lt b, then check the nature of roots of the equation (a -b)^(2) x^(2) + 2(a+ b - 2c)x + 1 = 0

sin^(-1) ""(2a)/(1+a^(2))-cos^(-1) ""(1-b^(2))/(1+b^(2))=2tan ^(-1) ""(a-b)/(1+ab)

If sin A+ tan A= a and cos A+ cot A= b, then show that (1+a)^(-2)+(1+b)^(-2)=(1-ab)^(-2)

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