Subtract and simplify?

Question

(sinx-1/cosx)-(cosx/sinx-1)

please show steps. it should be simplified to 2tanx, but i do not know how to reach that answer. please show steps

Answer 1

(sinx - 1) / cosx - (cosx / (sinx - 1)

Subtract one fraction from the other as normally done,
for example, a / b - c / d = (ad - bc) / (bd).

= [(sinx - 1)^2 - cos^2x] / [cosx(sinx - 1)]

Expand the numerator :
= (sin^2x - 2sinx + 1 - cos^2x) / [cosx(sinx - 1)]

From the relation, sin^2x + cos^2x = 1,
we get sin^2x = 1 - cos^2x

So, substituting sin^2x where 1 - cos^2x is,
in the numerator, gives :
(sin^2x - 2sinx + sin^2x) / [cosx(sinx - 1)]

Adding gives :
(2sin^2x - 2sinx) / [cosx(sinx - 1)]

Factorising gives :
2sinx(sinx - 1) / [cosx(sinx - 1)]

Cancelling out (sinx - 1) gives :
2sinx / cosx

= 2tanx