Every time you're dealing with a function that is actually the quotient of two other functions, let's call them f(x) and g(x), you can differentiate said function by using the quotient rule.
For y = f(x)/g(x), you have
color(blue)(d/dx(y) = (f^'(x) * g(x) - f(x) * g^'(x))/[g(x)]^2, where g(x) !=0
In your case, you have
y = sinx/(1 - cosx), with
{(f(x) = sinx), (g(x) = 1 - cosx) :}
You alsoe need to remember that
d/dx(sinx) = cosx
and that
d/dx(cosx) = -sinx
So, the derivatives of f(x) and g(x) will be
d/dx(f(x)) = d/dx(sinx) = cosx
d/dx(g(x)) = d/dx(1 - cosx)
d/dx(g(x)) = d/dx(1) - d/dx(cosx) = 0 - (-sinx) = sinx
The derivative of your function y will thus be
d/dx(y) = y^' = (cosx * (1 - cosx) - sinx * sinx)/(1 - cosx)^2
y^' = (cosx - cos^2x - sin^2x)/(1 - cosx)^2
y^' = (cosx - (sin^2x + cos^2x))/(1 - cosx)^2
You can simplify this further by using the fact that
color(blue)(sin^2x + cos^2x = 1
to get
y^' = color(green)((cosx - 1)/(1 - cosx)^2)
