Complete this equation Sin (x+y) / sin (x-y)

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