If `x` is very large as compare to `y ,` then prove that `sqrt(x/(x+y))dotsqrt(x/(x-y))=1+(y^2)/(2x^2)` .

Simplify (x^2)/(x+y) -(y^2)/(y+x)

If y=sqrt(x+1)+sqrt(x-1) , prove that sqrt(x^(2)-1)(dy)/(dx)=1/2y

y=sin^(-1)[sqrt(x-a x)-sqrt(a-a x)] then prove that 1/(2sqrtx sqrt(1-x))

If x + y + z = xyz, then prove that (2x)/(1- x^(2)) + (2y)/(1 - y^(2)) + (2z)/(1 - z^(2)) = (2x)/(1 - x^(2))cdot (2y)/(1 - y^(2))cdot (2z)/( 1- z^(2)) .

x dy - y dx = sqrt(x^(2) + y^(2)) dx

Prove that |(1,x,x^2),(1,y,y^2),(1,z,z^2)|=(x-y)(y-z)(z-x)

The solution of (x dx+y dy)/(x dy-y dx)=sqrt((1-x^2-y^2)/(x^2+y^2)) is

(a+bx)e^(y/x)=x , Prove that x^3(d^2y)/(dx^2)=(x(dy)/(dx)-y)^2

If y= 2^((1)/(log_(x)4)) then prove that x=y^(2) .

If x_1,x_2,x_3 as well as y_1, y_2, y_3 are in G.P. with same common ratio, then prove that the points (x_1, y_1),(x_2,y_2),a n d(x_3, y_3) are collinear.