Question

How do you simplify `(sec x - cos x) / tan x`?

Answer 1

`sinx`

This can also be proven by showing that `secx-cosx = (tanx)(sinx)` and then dividing both sides by `tanx`

Explanation:

Let's start by breaking down some terms. In my opinion, you have to kind of play around with trig stuff to get it to break down right.

`secx=1/cosx=tanx/sinx`

So,

`(secx-cosx)/tanx = secx/tanx - cosx/tanx = (tanx/sinx)/tanx - cosx/tanx`

`=1/sinx - cosx/tanx`

Tangent = sine/cosine, so the reciprocal of the tangent = cosine/sine

`= 1/sinx - cos^2x/sinx = (1-cos^2x)/sinx`

Since `sin^2x+cos^2x=1`, that means `cos^2x=1-sin^2x`

`= (1-(1-sin^2x))/sinx = (1 - 1 + sin^2x)/sinx = sin^2x/sinx = sinx`

Final Answer