If `tan^(-1)sqrt((a^(2)-x^(2))/(a^(2)+x^(2))`+`tan^(-1)sqrt((b^(2)-y^(2))/(b^(2)+y^(2))`=`(alpha)/(2)`, then show that,
`(x^(4))/(a^(4))-2(x^(2)y^(2))/(a^(2)b^(2))cos alpha+(y^(4))/(b^(4))=sin^(2)alpha`

if cos ^(-1 ) ""(x )/(a) + cos ^(-1 )""(y)/(b ) = alpha prove that , (x^(2) )/(a^(2)) -(2xy)/(ab ) ""cos alpha +(y^(2))/(b^(2))= sin ^(2) alpha

If x/y = (a + 2)/(a - 2) , then show that (x^(2) - y^(2))/(x^(2) + y^(2)) = (4a)/(a^(2) + 4) .

"If "y=(2)/(sqrt(a^(2)-b^(2))){tan^(-1)(sqrt((a-b)/(a+b))tan""(x)/(2))}," then show that "(d^(2)y)/(dx^(2))=(b sin x)/((a+b cos x)^(2)).

If x/y= (a+2)/(a-2) show that (x^2-y^2)/(x^2+y^2)=(4a)/(a^2+4)

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

x=sqrt(a^(2)cos^(2)alpha+b^(2)sin^(2)alpha)+sqrt(alpha^(2)sin^(2)alpha+b^(2)cos^(2)alpha) then x^(2)=alpha^(2)+b^(2)+2sqrt(p(a^(2)+b^(2))-p^(2)) , where p is equal to

If x^2+y^2 alpha xy then show that x+y alpha x-y .

If y="tan"^(-1) (sqrt(1+x^(2))-sqrt(1-x^(2)))/(sqrt(1+x^(2))+sqrt(1-x^(2))) show that, (dy)/(dx)=(x)/(sqrt(1-x^(4)))

If "sin"^(-1) (x)/(a)+"sin"^(-1)(y)/(b)="sin"^(-1)(c^(2))/(ab) , prove that, b^(2)x^(2)+2xy sqrt(a^(2)b^(2)-c^(4))+a^(2)y^(2)=c^(4)

If (sqrt(a + 2b) + sqrt(a - 2b))/(sqrt(a + 2b) - sqrt(a - 2b)) = sqrt3 and a^(2) + b^(2) = 1 , then the find values of a and b. (b) If sqrt((x - sqrt(a^(2) - b^(2)))^(2) + y^(2)) + sqrt((x + sqrt(a^(2) - b^(2)))^(2) + y^(2)) = 2a then prove that (x^(2))/(a^(2)) + (y^(2))/(b^(2)) = 1