Question

Prove that sin²x = (x² + y²)/2xy only when x = y

Answer 1

LHS=sin^2x
RHS=(x^2+y^2)/2xy
but x=y
so , (x^2+x^2)/2x^2=2x^2/2x^2=1
also we know
sinx €[-1,1]
also,
sin^2x€[0,1]
it means sin^2x largest value is 1
so from this statement above identity is true

sin^2x=1 =(x^2+y^2)/2xy
when x=y
it is not possible x> y, and x <y
because from
AP and Gp theory
Ap>GP
( x^2+y^2)/2>(x^2.y^2) ^1/2
(x^2+y^2)>2xy
(x^2+y^2)/2xy>1
but sin^2xlargest value is 1 hence sin^2x=(x^2+y^2)/2xy possible only when x=y