Question

if tan a +sin a= x
tan a -sin a= y
show x square - y square=4 rootxy

Answer 1

x^2 = tan^2 +2 tan sin + sin^2
y^2 = tan^2 -2 tan sin + sin^2

x^2-y^2 = 4 tan sin = 4 sin^2/cos
____________________________________
x y = tan^2 -sin^2 = sin^2/cos^2 -sin^2

= sin^2/cos^2 - sin^2 cos^2/cos^2

= [sin^2 / cos^2] [1-cos^2]
but 1 - cos^2 = sin^2
so
= sin^4/cos^2
so
sqrt(xy) = sin^2/cos
and sure enough
4 sqrt (xy) = 4 sin^2/cos

Answer 2

or, try this to start out

x+y = 2tan
x-y = 2sin
x^2-y^2 = (x+y)(x-y) = 4tan sin = 4sin^2/cos

Saves having to expand those pesky squared binomials.

Answer 3

but much more fun my way :)