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)
=(sinx-1)(sinx-1)-cos^2x/{cosx(sinx-1)
=(sinx-1)^2-cos^2 x)/{cosx(sinx-1)
=(sin^2x-2sinx+1-cos^2 x)/[cosx(sinx-1)
=sin^2x-2sinx+sin^2x+cos^2 x-cos^2 x)/{cox(sinx-1)
=(2sin^2x-2sinx)/{coxx(sinx-1)
=2sinx(sinx-1)/{cosx(sinx-1)
=2sinx/cosx
=2tanx

Answer 2

step 1 : subtract as normal fraction

((sinx-1)^2 - cosx^2)/( (cosx) (sinx-1) )

Step 2 : expand (sinx-1)^2

(sinx^2 - 2.sinx + 1 - cosx^2)/( (cosx) (sinx-1) )

Step 3 : replace 1 - cosx^2 with sinx^2

(2sinx^2 - 2sinx)/( (cosx) (sinx-1) )

Step 4 : factorize

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

Step 5 : Take out the common factor

2 sinx / cosx

Step 6 :

2 tanx

NOTE : This is only true when sinx != 1, i,e x <> 90 degree.

Answer 3

((sinx - 1)/cosx) - (cosx/(sinx-1)) =
((sinx - 1)^2 - cos^2x) / cosx(sinx - 1) =
(sin^2x - 2sinx + 1 - cos^2x) / cosx(sinx - 1) =
(2sin^2x - 2sinx) / cosx(sinx - 1) =
2sinx(sinx - 1) / cosx(sinx - 1) =
2sinx / cosx =
2tanx