If cot x=(2n^2+2mn)/(m^2+2mn) then prove that
csc^2x=1+((2n^2+2mn)/(m^2+2mn))^2 or
csc^2x=((m^2+2mn)^2+(2n^2+2mn)^2)/(m^2+2mn)^2
= ((m^4+4m^2n^2+4m^3n)+(4n^4+4m^2n^2+8mn^3))/(m^2+2mn)^2
(using identity (a+b)^2=a^2+b^2+2ab)) and now adding and rearranging terms
csc^2x=((m^4+4m^3n+8m^2n^2+8mn^3+4n^4))/(m^2+2mn)^2 ...(A)
Now (m^2+2mn+2n^2)^2 becomes
(m^4+4m^2n^2+4n^4+2*m^2*2mn+2*m^2*2n^2+2*2mn*2n^2)
(using identity (a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc)
or (m^4+4m^2n^2+4n^4+4m^3n+4m^2n^2+8mn^3)
or (m^4+4m^3n+8m^2n^2+8mn^3+4n^4) ....(B)
Note that in numerator of (A) and in (B), we have arranged monomials in decreasing degrees of m and increasing degree of n. Also note that both are equal.
Hence, csc^2x=((m^2+2mn+2n^2)^2)/((m^2+2mn)^2)
or cscx=(m^2+2mn+2n^2)/(m^2+2mn)
