Find an identity for \(\cos(4\theta)\) in terms of \(\sin(\theta)\) and \(\cos(\theta)\).
Solution
Write \(x=\cos(\theta), y= \sin(\theta)\), so \(e^{i\theta} = x+iy\). Then \[ \cos(4\theta) + i\sin(4\theta) &= e^{4i\theta } \\ &= (x+iy)^4 \\ &= \sum_{0\leq k \leq 4} {4\choose k} x^k (iy)^{4-k} \\ &= x^4 + 4ix^3y - 6x^2y^2 - 4ixy^3 + y^4 \\ &= (x^4 - 6x^2y^2 + y^4) + i(4x^3y - 4xy^3) .\] So \[ \cos(4\theta) &= \cos^4(\theta) - 6\cos^2(\theta)\sin^2(\theta) + \sin^4(\theta)\\ \sin(4\theta) &= 4\cos^3(\theta)\sin(\theta) - 4\cos(\theta)\sin^3(\theta) .\]