How do I verify the identity  `(cot^2 x -1) /(1+cot^2 x) = 1-2sin^2 x` ``

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

We know that: `cotx = cosx/sinx`

`==> ((cos^2 x)/(sin^2 x) -1)/(1+(cos^2 x)/(sin^2 x)) = 1- 2sin^2 x`

`Now we will use common denominator.`

`==> ((cos^2 x - sin^2 x)sin^2 x)/((sin^2 x+cos^2 x)(sin^2 x)) = 1- 2sin^2 x`

Now we will reduce `sin^2 x ` and substitute `sin^2 x + cos^2 x = 1`

`==> cos^2 x - sin^2 x = 1- 2sin^2x`

Now we know that `sin^2 x + cos^2 x = 1 ==> cos^2 x = 1- sin^2 x.`

`==> (1-sin^2 x) - sin^2 x = 1- 2sin^2 x`

`==> 1- 2sin^2 x = 1- 2sin^2 x`

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