If `x=tan t` and `y=tan pt`, prove that,
`(1+x^(2))y_(2)+2(x-py)y_(1)=0`

If x=a sin theta and y=b tan theta , then prove that (a^(2))/(x^(2))-(b^(2))/(y^(2))=1 .

If sinalpha=x and tan alpha=y, prove that, 1/x^2-1/y^2=1 .

If y=tan^-1x , prove that , (x^2+1)(d^2y)/(dx^2)+2xdy/dx=0 .

If y=e^x , Prove that , xy_2-(2x-1)y_1+(x-1)y=0 .

If y= (tan^(-1)x)^(2) , show that (x^(2)+1)^(2) y_(2)+2x(x^(2)+1)y_(1)=2 .

Given, y=tan^-1sqrt(x^2-1), if y_1 and y_2 are the first and second derivatives of y, then prove that x(x^2-1)y_2+(2x^2-1)y_1=0 .

If y=(tan^-1x)^2 , show that, (1+x^2)^2(d^2y)/(dx^2)+2x(1+x^2)dy/dx-2=0 .

If y=log (1+ sin x) , prove that y_(4)+y_(3)y_(1)+y_(2)^(2)=0 .

if tan ^(-1) x+tan ^(-1) y+tan ^(-1) z=pi prove that x+y+z=xyz

If x/(tan (theta + alpha)) = y/(tan (theta + beta)) = z/tan(theta + gamma), "prove that" (x+y)/(x-y)sin^(2) (alpha - beta) + (y + z)/(y - z) sin^(2) (beta - gamma) + (z + x)/(z-x) sin^(2) (gamma - alpha ) = 0