Given x+y=pi, evaluate the ratio (cosx+i*sinx)/(cosy-i*siny)?

We have to find the value of (cos x + i*sin x)/(cos y -
i*sin y) given that x + y = pi

(cos x + i*sin x)/(cos y -
i*sin y)

=> (cos x + i*sin x)(cos y + i*sin y)/(cos
y - i*sin y)(cos y + i*sin y)

=> (cos x + i*sin
x)(cos y + i*sin y)/((cos y)^2 + (sin y)^2)

=> cos
x*cos y + i*sin x*cos y + i*cos x*sin y +i^2*sin x*sin
y

=> cos x*cos y - sin x*sin y + i*(sin x*cos y +
cos x*sin y)

=> cos (x + y) + i*(sin x +
y)

=> cos pi + i*sin
pi

=> -1 +
i*0

=>
-1

The ratio is equal to
-1.