Question

What is the derivative of `y = secx/tanx`?

Answer 1

`y' = -cotxcscx`

Explanation:

Let's rewrite the function `y = secx/tanx` in sine and cosine.

`y = secx/tanx`

Apply the identities `sectheta = 1/costheta` and `tantheta = sintheta/costheta`.

`y = (1/cosx)/(sinx/cosx)`

`y = 1/cosx xx cosx/sinx`

`y = 1/sinx`

We can now use the quotient rule to differentiate.

`y' = (0 xx sinx - cosx xx 1)/(sinx)^2`

`y' = -cosx/sin^2x`

`y' = -cosx/sinx xx 1/sinx`

Apply the identities `cosx/sinx = cotx` and `1/sinx = cscx`.

`y' = -cotxcscx`

Hopefully this helps!