Question

\(1+{ }^{n} C_{1} \cos \theta+{ }^{n} C_{2} \cos 2 \theta+\ldots \ldots+{ }^{n} C_{n} \cos n \theta\) equals
(A) \(\left(2 \cos \frac{\theta}{2}\right)^{n} \cos \frac{n \theta}{2}\)
(B) \(2 \cos ^{2} \frac{n \theta}{2}\)
(C) \(2 \cos ^{2 n} \frac{\theta}{2}\)
(D) \(\left(2 \cos ^{2} \frac{\theta}{2}\right)^{n}\)

Answer 1

Ans(A)

 \(\operatorname{Re}\left({ }^{n} C_{0}+{ }^{n} C_{1} e^{i \theta}+\ldots . .\right)=\operatorname{Re}\left(1+e^{i \theta}\right)^{n}=\operatorname{Re}(\cos \theta+1+i \sin \theta)^{n}=(2 \cos (\theta / 2))^{n} \cos \left(\frac{n \theta}{2}\right)\)