Answer to Question #114737 in Algebra for ANJU JAYACHANDRAN

Question

Express sin^x as a linear combination of sin kx and cos kx, k element of Z.

Answer 1

I suppose that the task lacks the degree n of the sine, i.e.: sinⁿx.

For even exponents (i.e., for n=2, 4, 6, …) 

$sin^nx =\frac {C^\frac n 2_n}{2^n}+\frac 1 {2^{n-1}}*\displaystyle\sum_{k=0}^{\frac n 2-1}(-1)^{\frac n 2-k}*C^k_n*cos((n-2*k)x)$

for odd exponents (i.e., for n=3, 5, 7, …) 

$sin^nx =\frac 1 {2^{n-1}}*\displaystyle\sum_{k=0}^{\frac {n-1} 2}(-1)^{\frac {n-1} 2-k}*C^k_n*sin((n-2*k)x)$

where $C^k_n= \frac {n!} {k!*(n-k)!}$ is the number of combinations of n elements by k.

Check with n=2:

$sin^2x =\frac {C^1_2}{4}+\frac 1 {2}*\displaystyle\sum_{k=0}^{0}(-1)*C^0_2*cos(2x)=\frac {1-cos(2x)} {2}$

$C^0_2= 1,$ $C^1_2 =2$

Or

$sin^2x = 1-cos^2x$ ; $cos(2x) = 2cos^2x-1\implies sin^2x = \frac {1-cos(2x)} {2}$