`|a| lt 1 and |b| lt 1, x=1 + a + a^(2) +`……. and `y=1 + b + b^(2)`+ …..,Prove that, `1 + ab + a^(2) b^(2) + ….= (xy)/(x+y-1)`

y = sqrt(1 + x^(2)) : y' = (xy)/(1 + x^(2))

If x, y in R and |{:((a ^(x) + a ^(-x) ) ^(2) , (a ^(x) - a ^(-x) ) ^(2), 1),((b ^(x) + b ^(-x)) ^(2),(b ^(x) - b ^(-x) ) ^(2) , 1), ( ( c ^(x) + c ^(-x)) ^(2) , (c ^(x) - c ^(-x)) ^(2) , 1):}|=2 y + 6 then y = ____________.

If y= (x + sqrt(x^(2) + 1))^(m) then prove that, (x^(2)+1) y_(2) + xy_(1)= m^(2)y

If a, b, c are in G.P. and a^(1/x) =b^(1/y)=c^(1/z) , prove that x, y , x are in A.P

If x=1+(log)_a b c , y=1+(log)_b c a and z=1+(log)_c a b , then prove that x y z=x y+y z+z x

Let a,b,c be in A.P. and |a|lt1,|b|lt1|c|lt1. if x=1+a+a^(2)+ . . . . oo , y=1+b+b^(2)+ . . . .oo , z=1+c+c^(2)+ . . . oo , then x,y,z are in

If y= a sin x + b cos x then, y^(2) + (y_(1))^(2) = ……. (a^(2) + b^(2) ne 0)

If y= sin (pt), x= sin t , then prove that (1- x^(2))y_(2)-xy_(1)+ p^(2)y= 0

If 2x= y^((1)/(m)) + y^(-(1)/(m)) (n ge 1) then prove that, (x^(2)-1) y_(2) + xy_(1) = m^(2)y