Question

The value of $(1+\cos \theta +i\sin \theta {)}^n+(1+\cos \theta -i\sin \theta {)}^n$ is

• A. $2^n.{\cos }^n\frac{\theta }{2}.\cos \frac{n\theta }{2}$
• B. $2^{n+1}.{\cos }^n\frac{\theta }{2}.\cos \frac{\theta }{2}$
• C. $2^{n+1}.{\cos }^n\frac{\theta }{2}.\cos \frac{n\theta }{2}$
• D. $2^n.{\cos }^n\frac{\theta }{2}.\cos \frac{\theta }{2}$

Answer 1

The correct option is C $2^{n+1}.{\cos }^n\frac{\theta }{2}.\cos \frac{n\theta }{2}$

Using,
$1+\cos \theta =2{\cos }^2\left(\frac{\theta }{2}\right),$
$\sin \theta =2\sin \left(\frac{\theta }{2}\right)\cos \left(\frac{\theta }{2}\right)$
$(1+\cos \theta +i\sin \theta {)}^n+(1+\cos \theta -i\sin \theta {)}^n$
$={(2{\cos }^2\frac{\theta }{2}+i\cdot 2\sin \frac{\theta }{2}\cos \frac{\theta }{2})}^n+{(2{\cos }^2\frac{\theta }{2}-i\cdot 2\sin \frac{\theta }{2}\cos \frac{\theta }{2})}^n$
$=2^n{\cos }^n\left(\frac{\theta }{2}\right)(\cos \frac{n\theta }{2}+i\sin \frac{n\theta }{2})+2^n{\cos }^n\frac{\theta }{2}(\cos \frac{n\theta }{2}-i\sin \frac{n\theta }{2})$
$=2^{n+1}{\cos }^n\left(\frac{\theta }{2}\right)\left(\cos \frac{n\theta }{2}\right)$