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