Prove that `""^(2n+1)P_(n-1)=((2n+1)!)/((n+2)!)` and `""^(2n-1)P_n=((2n-1)!)/((n-1)!)`

Prove that ((n+1)/(2))^(n) gt (n!)

Prove that cos^(-1)((1-x^(2n))/(1+x^(2n)))=2 tan^(-1)x^n

If a , a_(1) , a_(2) ,a_(3) , cdots a_(2n), b are in A.P. and a, g_(1) , g_(2) , g_(3) , cdots , g_(2n), b are in G.P. and h is the H.M of a and b then prove that (a_(1)+a_(2n))/(g_(1)g_(2n))+(a_(2)+a_(2n-1))/(g_(2)g_(2n-1))+cdots+ (a_(n)+a_(n+1))/(g_(n)g_(n+1))=(2n)/(h)

Prove that ""^(n)C_(0)""^(n)C_(0)-^(n+1)C_(1) ""^(n)C_(1)+^(n+2)C_(2)""^(n)C_(2)....=(-1)^(n)

if ^(2n+1)p_(n-1):^(2n-1)p_n=3:5 , prove that n = 4

Prove that sin(n+1)xsin(n+2)x+cos(n+1)xcos(n+2)x=cosx

(1+ (C_1)/(C_0)) (1+(C_2)/(C_3) )....(1+(C_n)/(C_(n-1))) is equal to : a) (n+1)/(n!) b) ((n+1)^(n))/((n-1)!) c) ((n-1)^(n))/(n!) d) ((n+1)^(n))/(n!)

If a_(1), a_(2),a_(3), cdots , a_(2n+1) are in A.P then (a_(2n+1)-a_(1))/(a_(2n+1)+a_(1)) + (a_(2n)-a_(2))/(a_(2n)+a_(2))+cdots+ (a_(n+_2)-a_(n))/(a_(n+2)+a_(n)) is equal to

Prove that 1+(1)/(4)+(1)/(9)+...........+(1)/(n^(2))lt2-(1)/n,(ngt1)