I encountered the following approximation for $\sin(x)$ in a physics book:
$$ \textrm{for }x\in[0,\pi]\textrm{, }\quad \sin^3(x)\approx \left[\frac{\cos\left(\frac{\pi}{2}\cos(x)\right)}{\sin(x)}\right]^2 $$ Plotting both of these functions on the specified interval produces the following Mathematica Plot, indicating that the approximation is quite good. I was wondering if there was a way to get an intuition for where this approximation comes from or even a rigorous derivation. Here's what I've attempted so far:
Because there's a half in the argument of the outer cosine, I tried using the half angle Identity to simplify the RHS of the approximation and got $$ \frac{\left(\cos\left(\pi\cos\left(x\right)\right)+1\right)}{2\sin^{2}\left(x\right)}$$ Because both functions are equal at $x=\frac{\pi}{2}$ I tried using a Maclaurin series in $u=x-\frac{\pi}{2}$. However, the coefficients never match because the LHS always has rational coefficients whereas the RHS has coefficients which are rational multiples of powers of $\pi$.
I then tried expanding either side in terms of Legendre polynomials in u (renormalized to be orthonormal with the inner product $(L_m,L_n)=\int_{-\frac{\pi}{2}} ^{\frac{\pi}{2}}L_mL_ndu$ and got the following values: $$n=0: 0.447, 0.424$$ $$n=2: −0.736, −0.745$$ $$n=4: 0.376, 0.427$$ $$n=6: −0.110, −0.122$$ With the first value corresponding to the coefficient for the RHS and the second value corresponding to the coefficient for the LHS. These coefficients don't get closer for larger values of $n$ as far as I could tell, so this method failed as well, indicating that this approximation didn't come from messing around with Legendre polynomials (I'm assuming that other sets of orthogonal polynomials will fail as well).
Any insight on this problem would be greatly appreciated!