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\frac{\sqrt{2}\left(\cos(\frac{8x+\pi }{4})+\sqrt{2}\left(\sin(x)\right)^{2}\right)}{2}
Factor out \frac{1}{2}.
\sqrt{2}\left(\cos(\frac{8x+\pi }{4})+\sqrt{2}\left(\sin(x)\right)^{2}\right)
Consider \sqrt{2}\left(\cos(\frac{1}{4}\left(8x+\pi \right))+\sqrt{2}\left(\sin(x)\right)^{2}\right). Factor out \sqrt{2}.
\frac{\sqrt{2}\left(\cos(\frac{8x+\pi }{4})+\sqrt{2}\left(\sin(x)\right)^{2}\right)}{2}
Rewrite the complete factored expression.