Let `k=1^@`, then prove that `sum_(n=0)^88 1/(cosnk* cos(n+1)k)=cosk/sin^2k`

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

if omega is the nth root of unity and Z_1 , Z_2 are any two complex numbers , then prove that . Sigma_(k=0)^(n-1)| z_1+ omega^k z_2|^2=n{|z_1|^2+|z_2|^2} where n in N

Evaluate sum_(k-1)^11(2+3^k)

A=[{:(a,b),(0,1):}],ane1 then prove that A^(n)=[{:(a^(n),(b(a^(n)-1))/(a-1)),(0," "1):}], n inN .

sin(n+1)x.cosnx-cos(n+1)x.sinnx= …………

The value of lim_(n->oo) sum_(k=1)^n log(1+k/n)^(1/n) ,is

Prove by mathematical induction that sum_(r=0)^(n)r^(n)C_(r)=n.2^(n-1), forall n in N .

Definite integration as the limit of a sum : lim_(ntooo)sum_(k=0)^(n)(n)/(n^(2)+k^(2))=..........

If cosec theta + cot theta =k , then prove that cos theta =(k^(2)-1)/( k^(2)+1)

There are n straight lines in a plane such that n_(1) of them are parallel in onne direction, n_(2) are parallel in different direction and so on, n_(k) are parallel in another direction such that n_(1)+n_(2)+ . .+n_(k)=n . Also, no three of the given lines meet at a point. prove that the total number of points of intersection is (1)/(2){n^(2)-sum_(r=1)^(k)n_(r)^(2)} .