If `u_n=sin^("n")theta+cos^ntheta,` then prove that `(u_5-u_7)/(u_3-u_5)=(u_3)/(u_1)` .

If U_n=(sqrt(3)+1)^(2n)+(sqrt(3)-1)^(2n) , then prove that U_(n+1)=8U_n-4U_(n-1)dot

If u = sin ((x)/(y)) prove that u _(xy) = u _(yz).

If u = cos ^(-1) ((x)/(y)) prove that x (del u)/(delx) + y (del u)/(dely)=0.

If u(x,y) = (x^(2) + y^(2))/sqrt(x+y) , prove that x(del u)/(del x) + y(del u)/(del y) = 3/2 u .

If z_1a n dz_2 are complex numbers and u=sqrt(z_1z_2) , then prove that |z_1|+|z_2|=|(z_1+z_2)/2+u|+|(z_1+z_2)/2-u|

If u=sin^(-1)((x+y)/(sqrtx+sqrty)) show that x(delu)/(delx)+y(delu)/(dely)=1/2 tan u.

If u= tan^(-1) ((x^(3) + y^(3))/(x-y)) , Prove that x(del u)/(del x) + y(del u)/(del y) = sin 2u .