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

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

Find the value of sum_(i=0)^(oo)sum_(j=0)^(oo)sum_(k=0)^(oo)(1)/(3^(i)3^(j)3^(k)) .

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

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

sin(n+1)x.cosnx-cos(n+1)x.sinnx= …………

The value of the determinants |{:(1,a,a^(2)),(cos(n-1)x,cos nx , cos(n+1)x),(sin(n-1)x , sin nx , sin(n+1)x):}| is zero if

Find the value of sum_(k=1)^10[sin((2pik)/(11))-icos((2pik)/(11))],wherei=sqrt(-1).

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

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