Prove that `Sigma_(K=0)^(n) ""^nC_(k)` sin K x , cos (n-K)x =`2^(n-1)` sin x.

sin(n+1)x.cosnx-cos(n+1)x.sinnx= …………

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

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

Prove that cos theta cos 2theta cos2^2 theta … cos 2^(n-1) theta = (sin2^n theta)/(2^n.sin theta) , for all n in N

Prove that , sin theta + sin 2theta + ……+ sin n theta = ( (sin( n theta/(2)) sin ((n+1)/(2))theta))/(sin theta/2) , for all n in N .

((n),(r)) = (""^(n)P_(r))/(K), then K = ......

The value of 5 * cot ( Sigma_(k =1)^(5) cot ^(-1) ( k^(2) + k + 1)) is equal to

If the range of y=sin^(-1)x+cos^(-1)x+tan^(-1) x is [k,K] , then

Given that bar(x) is the mean and sigma^(2) is the variance of n observation x_(1), x_(2), …x_(n). Prove that the mean and sigma^(2) is the variance of n observations ax_(1),ax_(2), ax_(3),….ax_(n) are abar(x) and a^(2)sigma^(2) , respectively, (ane0) .