Question

State true or false:

${\cos }^n\alpha -n{\cos }^{n-1}\alpha \cos \alpha +\frac{n(n-1)}{2!}{\cos }^{n-2}\alpha \cos 2\alpha +\cdots (n+1)$ terms$={(-1)}^{n/2}{\sin }^n\alpha$ , when n is even,$=0$ when n is odd. Type 1 for true and 0 for false

Answer 1

The given series is cosins series as it has multiple angles with $\cos .$ For convenience sake put $\cos \alpha =a$
$\therefore C=a^n-n{a}^{n-1}\cos \alpha +\frac{n(n-1)}{2!}{a}^{n-2}\cos 2\alpha +\cdots$
$S=-n{a}^{n-1}\sin \alpha +\frac{n(n-1)}{2!}{a}^{n-2}\sin 2\alpha +\cdots$
Note: We have changed cosines of multiple angles with sines of multiple angles only.
$\therefore C+iS=a^n-n{a}^{n-1}{e}^{ia}+\frac{n(n-1)}{2!}{a}^{n-2}{e}^{2i\alpha }+\cdots$
C+iS is of the form of Binomial expansioln
$={(a-e^{i\alpha })}^n={[\cos \alpha -(\cos \alpha +i\sin \alpha )]}^n$
$={(-1)}^n{i}^n{\sin }^n\alpha ={(-1)}^{n/2}{\sin }^n\alpha$ if n is even.
=purely imaginary if n is odd.
Equating real parts,
$\therefore C={(-1)}^{n/2}{\sin }^n\alpha ,$ when n is even
$=0$ when n is odd.