Question

How do you prove: `(1-2sin^2(x))`//`(sin(x))=csc(x)-sin(x)`?

Answer 1

The given equation does not appear to be true in general.
(So any "proof" would be faulty).
...Assuming the double slashes on the left side were a typo and that division was intended

Explanation:

We are asked to prove:
`color(white)("XXX")(1-2sin^2(x))/(sin(x))=csc(x)-sin(x)`

Consider the case `x=pi/2color(white)("xx")rarrcolor(white)("xx")sin(x)=1`

L.S.
`color(white)("XXX")=(1-2sin^2(x))/(sin(x))`

`color(white)("XXX")=(1- 2 *(1)^2)/1`

`color(white)("XXX")=-1`

R.S.
`color(white)("XXX")=csc(x)-sin(x)`

`color(white)("XXX")=1/(sin(x))-sin(x)`

`color(white)("XXX")=1/1-1`

`color(white)("XXX")=0`

Since L.S.`!=`R.S. the given equation can not be proven.