Prove that `sum_(r=0)^n^n C_rsinr xcos(n-r)x=2^(n-1)sin(n x)dot`

Prove that sum_(r=0)^ssum_(s=1)^n^n C_s^ s C_r=3^n-1.

Prove that sum_(r=0)^n 3^r n Cundersetr = 4^n .

Prove that sum_(r=0)^(2n)r(.^(2n)C_r)^2=n^(4n)C_(2n) .

If p+q=1, then show that sum_(r=0)^n r^2^n C_rp^r q^(n-r)=n p q+n^2p^2dot

Prove that sum_(r=0)^n^n C_r(-1)^r[i^r+i^(2r)+i^(3r)+i^(4r)]=2^n+2^(n/2+1)cos(npi//4),w h e r ei=sqrt(-1)dot

If x+y=1, prove that sum_(r=0)^n r* ^nC_r x^r y^(n-r)=nxdot

Find the sum sum_(r=0)^n^(n+r)C_r .

Prove that sum_(r=1)^n(1/(costheta+"cos"(2r+1)theta))=(sinntheta)/(2sintheta* costheta*cos (n+1)theta),(w h e r e n in N)dot

Find the sum of sum_(r=1)^n(r^n C_r)/(n C_(r-1) .

Prove that sum_(k=0)^(n) (-1)^(k).""^(3n)C_(k) = (-1)^(n). ""^(3n-1)C_(n)