I am looking for trig identities of the form
$\sin(a+b)^{2k} = \sum a_i \sin^{2 n_i}(b_i a) \cos^{2 m_i}(c_i a) \sin^{2 l_i}(d_i b) \cos^{2 j_i}(e_i b)$
where $k,n_i,m_i,l_i,j_i$ are integers $> -1$ and $a_i,b_i,c_i,d_i,e_i$ are rational numbers$>0$.
Im not sure they exist for all $k$.
For instance
$$\sin^2(\alpha+\beta) = \cos^2 \alpha + \cos^2\beta - 2 \cos\alpha\cos\beta \cos\left(\alpha+\beta\right)$$
and
$$\sin^2\left(\alpha+\beta\right) = \sin^2\alpha + \sin^2\beta + 2 \sin\alpha\sin\beta\cos\left( \alpha+\beta \right)$$
but neither of those are sums of squares.
In particular I wonder about such addition formulas that are sums of squares for
$$\sin(a+b)^2$$
and
$$\sin(a+b)^4$$
Or reasons why they can not exist.
For what $k$ do such identities exist ?
edit
For clarity $\sin^{*}$ means power, not iteration ofcourse.
Someone suggested using complex numbers, but I am not sure how that helps.
I tried setting $a=b,a+b =2x$
and then somehow using
$$\sin^2(2x) = 4 \cos^2(x) \sin^2(x) $$
and setting $a = 2b, a + b = 3x$
and
$$\sin(3x)^2 = \sin^2(x)(2 \cos(2x)+1)^2$$
but it did not work out.