Question

How do you prove: `(4sin^2x)/(2sinx+1)=2sinx-1`?

Answer 1

The problem is that this is not a valid trig identity

Explanation:

For example, if `x=0` then
LS
`color(white)("XXX")=(4sin^2(x))/(2sin(x)+1)=(4xx1^2)/(2xx1+1)=4/3`
and
RS
`color(white)("XXX")=2sin(x)-1=2xx1-1=1`

Clearly `LS!=RS`

Perhaps the numerator (LS) should have been `4sin^2(x)-1`
(in which case the solution is fairly simple since `4sin^2(x)-1` can be factored as `(2sin(x)-1)(2sin(x)+1)`).