If x>y prove that `sqrt(y+sqrt(2xy-x^2))+sqrt(y-sqrt(2xy-x^2)) = sqrt(2x)`.

Simplify : (a) sqrt(y+sqrt(2xy-x^(2))) + sqrt(y-sqrt(2xy-x^(2))) (b) (x+sqrt(x^2-1))/(x-sqrt(x^(2)-1)) -(x-sqrt(x^(2)-1))/(x+sqrt(x^(2)-1))

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) .

Prove that sqrtx/(sqrtx +sqrty) + sqrty/(sqrtx-sqrty) + (2sqrt(xy))/(x-y) = (sqrtx+sqrty)/(sqrtx-sqrty)

If x ,y in R satify the equation x^2+y^2-4x-2y+5=0, then the value of the expression [(sqrt(x)-sqrt(y))^2+4sqrt(x y)]//(x+sqrt(x y)) is a. sqrt(2)+1 b. (sqrt(2)+1)/2 c. (sqrt(2)-1)/2 d. (sqrt(2)+1)/(sqrt(2))

Length of the latus rectum of the parabola sqrt(x) +sqrt(y) = sqrt(a) is (a) a sqrt(2) (b) (a)/(sqrt(2)) (c) a (d) 2a

Prove that sqrt(x^2+2x+1)-sqrt(x^2-2x+1)={-2, x 1

Rationalise the denominator: (a) (1)/(root(3)(3) + root(3)(2)) , (b) (2)/(sqrt5 + sqrt3 + sqrt2) , (c) (x^(2))/(sqrt(x^(2) + y^(2)) - y) , (d) (1)/(sqrt6 + sqrt5 - sqrt11) (e) (sqrt(x + 2y) - sqrt(x -2y))/(sqrt(x + 2y) + sqrt(x - 2y)) , (f) (sqrt10 + sqrt5 - sqrt3)/(sqrt10 - sqrt5 + sqrt3)

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) .

If x+y = z , then prove that 1/(log_((sqrt(z)-sqrt(y)))(x)) + 1/(log_((sqrt(z)+sqrt(y)))(x)) = 1 .

If y=(sqrt(a+x)-sqrt(a-x))/(sqrt(a+x)+sqrt(a-x)) ,then (dy)/(dx) is equal to (a) (a y)/(xsqrt(a^2-x^2)) (b) (a y)/(sqrt(a^2-x^2)) (c) (a y)/(xsqrt(a^2-x^2)) (d) none of these