Trigonometric Power Formulas

Author

Eric W. Weisstein
January 2, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Trigonometry/TrigonometricPowerFormulas.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/TrigonometricPowerFormulas.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Sums

(sin^2)(x)==((Binomial[2 n, n])/(2^(2 n)))+((((-1))^n)/(2^(2 n-1)))(∑_(k=0)^(n-1))(((-1))^k) (Binomial[2 n, k]) cos(2 x (n-k))

(1/(2^(2n)))Binomial[2n,n]+((((-1))^n)/(2^(2n-1)))Sum[(((-1))^k)Binomial[2n,k]Cos[2(n-k)x],{k,0,n-1}]

FullSimplify[%,n∈Integers&&n>0]

Table[%,{n,5}]//FullSimplify

((((-1))^n)/(4^n)) (∑_(k=0)^n)(((-1))^k) (Binomial[2 n+1, k]) sin(x (-2 k+2 n+1))==(sin^(2 n+1))(x)

((((-1))^n)/(4^n))Sum[(((-1))^k)Binomial[2n+1,k]Sin[(2n+1-2k)x],{k,0,n}]

FullSimplify[%,n∈Integers]

(1/(2^(2 n-1)))(∑_(k=0)^(n-1))(Binomial[2 n, k]) cos(2 x (n-k))+(1/(2^(2 n)))(Binomial[2 n, n])==(cos^(2 n))(x)

(1/(2^(2n)))Binomial[2n,n]+(1/(2^(2n-1)))Sum[Binomial[2n,k]Cos[2(n-k)x],{k,0,n-1}]

FullSimplify[%,n∈Integers&&n>0]

Table[%,{n,5}]//FullSimplify

(1/(4^n))(∑_(k=0)^n)(Binomial[2 n+1, k]) cos(x (-2 k+2 n+1))==(cos^(2 n+1))(x)

(1/(4^n))Sum[Binomial[2n+1,k]Cos[(2n+1-2k)x],{k,0,n}]