2008/008) Tough inequality

Prove that for all x>0, y>0, and all real a it holds true that

(x^((sin(a))^2))*(y^((cos(a))^2))
let us assume y > x

x^t y^(1-t) < x+ y putting (sin a)^2 = t we get (cos a)^2 = 1-t

devide by x on both sides

x^(t-1)y^(1-t) < (1+y/x)

or (y/x)^(1-t) < 1+ y/x

or m ^k < 1+m where k < 1 and m > 1
we know m ^k < m where k < 1

so m^ k < 1+m

if x > y then role of sin a and cos a are reversed and you get the same result