can somebody differentiate this and simplify?

Question

y= (1-sec x) / tan x

Answer 1

y = (1 - 1/cos x) / (sin x / cos x)
y = [(cos x - 1) / (cos x)] [ (cosx ) / sin x ]
y = [ cos x - 1] / sin x
y = cot x - cosec x
dy/dx = - cosec ² x + cosec x cot x
dy/dx = (cosec x) (cot x - cosec x)

Answer 2

Do the quotient rule. The cosines will cancel and you will be left with -sin x. I did not need to use trig identities, just the quotient rule. I did need to simplify tangent and secant to the sin x , cos x identities. Remember, sec x = 1/cos x and tan x = sin x/ cosx.