How to you develop a formula for sin(11`theta) as a degree- 11 polynomial in cos(`theta).
$\displaystyle \sin (11\theta) = \sin (10\theta + \theta) = \sin(10\theta)\cos(\theta) + \sin(\theta)\cos(10\theta) $
$\displaystyle = 2\sin(5\theta)\cos(5\theta)\cos(\theta) + \sin(\theta)[2\cos^2(5\theta) - 1]$
just continue the process of using the identities until the angle is $\displaystyle \theta$ only and in terms of $\displaystyle \cos \theta$