How verify the identity?  (4(sinx)^2 -1)/(2(sin x)^2 +1)= 2 sin x -1 

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I think that the content of the problem is not quite well written, hence, I suggest you to consider the denominator `2sin x + 1` and not `2sin^2 x + 1` , or else, the expression does not represent an identity.

Hence, the identity you need to check is now `(4(sinx)^2 -1)/(2(sin x) +1)= 2 sin x -1.`

You should convert the difference of squares from the numerator into a product, using the following formula, such that:

`a^2 - b^2 = (a - b)(a + b)`

Reasoning by analogy, yields:

`4(sinx)^2 -1 = (2sin x + 1)(2sin x - 1)`

Replacing the product for the difference of squares, yields:

`((2sin x + 1)(2sin x - 1))/(2(sin x) +1) = 2 sin x -1`

Reducing duplicate factors yields:

`2sin x - 1 = 2sin x - 1`

Hence, the last line proves to you that the identity ` (4(sinx)^2 -1)/(2(sin x) +1)= 2 sin x -1` holds.

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