SOLUTION: How to prove this following identity? (sin2x+cos2x)/(2cosx+sinx-2(cos^3x+sin^3x))=cosecx

Question 976230: How to prove this following identity?
(sin2x+cos2x)/(2cosx+sinx-2(cos^3x+sin^3x))=cosecx
Answer by FrankM(1040) (Show Source): You can put this solution on YOUR website!

First, the sum of cubes

This applies to the denominator

but you see the sin squared and cos squared add to 1?

simplify denominator a bit more to get:

Note that
and if we factor a sin out of the first two terms in denominator we get:

and can make the substitution:

Next we have

so let's factor sin out of the later part of denominator

and make the substitution

now, factor out sin in the denominator:

And after canceling numerator/denominator all we have left is

which is cosecant.
There may be a faster or more elegant method, but it's key to see that the sum of two cubes had to be factored to start the process.
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