Prove that: `sum_(m=1)^ntan^(-1)((2m)/(m^4+m^2+2))=tan^(-1)((n^2+n)/(n^2+n+2))`

Prove that : sum_(m=1)^n\ \ \ tan^(-1)((2m)/(m^4+m^2+2))=tan^(-1)((n^2+n)/(n^2+n+2))

sum_(m=1)^(n) tan^(-1) ((2m)/(m^(4) + m^(2) + 2)) is equal to

Prove that: tan^(-1)(m/n)+tan^(-1)((n-m)/(n+m))=pi/4

Prove that: tan^(-1)(m/n)+tan^(-1)((n-m)/(n+m))=[pi/4; m^(2)/n^(2) > -1

The sides of a triangel are such that a/(1+m^2+n^2)=b/(m^2+n^2)=c/((1-m^2)(1+n^2)) , prove that A=2tan^-1 m/n, B=2tan^-1 (mn) and /_\ = (mnbc)/(m^2+n^2)

If I_(m,n)= int(sinx)^(m)(cosx)^(n) dx then prove that I_(m,n) = ((sinx)^(m+1)(cosx)^(n-1))/(m+n) +(n-1)/(m+n). I_(m,n-2)

IfI_(m , n)=int_0^(pi/2)sin^m xcos^n xdx , Then show that I_(m , n)=(m-1)/(m+n)I_(m-2,n)(m ,n in N) Hence, prove that I_(m , n)=f(x)={((n-1)(n-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))pi/4 when both m and n are even ((m-1)(m-3)(m-5)(n-1)(n-3)(n-5))/((m+n)(m+n-2)(m+n-4))}

Prove that 1/(m !)^n C_0+n/((m+1)!)^n C_1+(n(n-1))/((m+2)!)^n C_2++(n(n-1)2xx1)/((m+n)!)^n C_n=((m+n+1)(m+n+2)(m+2n))/((m+n)!)

Evaluate sum_(m=1)^(oo)sum_(n=1)^(oo)(m^(2)n)/(3^(m)(n*3^(m)+m*3^(n))) .

For any positive integer (m,n) (with ngeqm ), Let ((n),(m)) =.^nC_m Prove that ((n),(m)) + 2((n-1),(m))+3((n-2),(m))+....+(n-m+1)((m),(m)) = ((n+2),(m+2))