help in trig please...?

Question

Need the full explination for the identity:


(Csc(x) - sin(x)) / cot^2 (x) = sin (x)

Please explain the process fully...

Answer 1

cscx-sin(x)=(1/sinx)-sinx
LCD is sinx
=(1-sin^2x)/sinx
=cos^2x/sinx
cot^2x=(cotx)^2
(cosx/sinx)^2
=cos^2x/sin^2x
now substituting
cscx-sinx/cost^2
(cos^2x/sinx)/(cos^2x/sin^2x)
inverting the (cos^2x/sin^2x)
and multiplying
(cos^x/sinx)(sin^2x/cos^2x)
cancelling
thisis=sinx

Answer 2

Again...
(1/sin - sin) / (cos^2/sin^2)
(sin - sin^3)/ cos^2 = sin (1 - sin^2)/cos^2

since cos^2 + sin^2 = 1 then cos^2 = 1 - sin^2

= sin cos^2/cos^2 = sin