Question

If n is an integer then, show that $(1+\cos \theta +i\sin \theta {)}^n+(1+\cos \theta -i\sin \theta {)}^n=2^{n+1}{\cos }^n\left(\frac{\theta }{2}\right)\cos \left(\frac{n\theta }{2}\right)$

Answer 1

$(1+\cos \theta +i\sin \theta {)}^n+(1+\cos \theta -i\sin \theta {)}^n=(2{\cos }^2\frac{\theta }{2}+i2\sin \frac{\theta }{2}\cos \frac{\theta }{2}{)}^n+(2{\cos }^2\frac{\theta }{2}-i2\sin \frac{\theta }{2}\cos \frac{\theta }{2}{)}^n$

$=2^n{\cos }^n\frac{\theta }{2}(e^{i\frac{n\theta }{2}}+e^{-i\frac{n\theta }{2}})=2^n{\cos }^n(\frac{\theta }{2})(2\cos \frac{n\theta }{2})=2^{n+1}{\cos }^n(\frac{\theta }{2})\cos (\frac{n\theta }{2})$