Answer:
Sin(x+y)/sin(x-y) = [ sin (x + y) ]^2 / (cos y)^2 - (cos x)^2
Step-by-step explanation:
Sin(x+y)/sin(x-y) = [sin x cos y + cos x sin y]/ [sin x cos y - cos x sin y]
[sin x cos y + cos x sin y]/ [sin x cos y - cos x sin y]
multiply top and bottom of this fraction by [sin x cos y + cos x sin y]
the denominator becomes:
( sin x cos y)^2 - (cos x sin y)^2
(sin y)^2 = 1 - (cos y)^2
( sin x cos y)^2 - (cos x sin y)^2
= ( sin x cos y)^2 - (cos x)^2 [ 1 - (cos y)^2 ]
= ( 1 - (cos x)^2) (cos y)^2 - (cos x)^2 [ 1 - (cos y)^2 ]
= (cos y)^2 - ((cos x)^2) (cos y)^2 - (cos x)^2 + [(cos x)^2] (cos y)^2
things cancel out
= (cos y)^2 - (cos x)^2
Sin(x+y)/sin(x-y) = [ sin (x + y) ]^2 / (cos y)^2 - (cos x)^2
