How do you simplify (cot^2 x)/(csc x +1)?

1 Answer
Truong-Son N.
May 5, 2016

The identity you're looking for is csc^2x - 1 = cot^2x.

You can derive this by starting from sin^2x + cos^2x = 1:

sin^2x + cos^2x = 1

cancel(sin^2x/sin^2x)^(1) + stackrel(cot^2x)overbrace(cos^2x/sin^2x) = stackrel(csc^2x)overbrace(1/sin^2x)

\mathbf(1 + cot^2x = csc^2x)

Thus:

color(blue)(cot^2x/(cscx + 1))

= (csc^2x - 1)/(cscx + 1)

= ((cscx - 1)cancel((cscx + 1)))/cancel((cscx + 1))

= color(blue)(cscx - 1)

...if and only if cscx + 1 ne 0.

If cscx + 1 = 0, then 1/sinx = - 1, which is the case when x = (3pi)/2 pm 2npi for all n in ZZ.

Therefore, this answer is valid when x ne (3pi)/2 pm 2npi for all n in ZZ.

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