Question

$\forall nϵN$, which of the following is true?

• A. $|\sin \left(nx\right)|<|\sin x|$
• B. $|\sin$ (nx) $|<n|\sin$$x|$
• C. $|\sin$ (nx) $|\le n|\sin$$x|$
• D. $\sin$ (nx) $\le sin\left(n\right)$

Answer 1

The correct option is B $|\sin$ (nx) $|<n|\sin$$x|$

Lets consider option B,

For $n=2$,

$|\sin \left(2x\right)|<2|\sin \left(x\right)|$ Which is true since

LHS$=$RHS$\times \cos x$

Given true for $n=k$, lets prove for $n=k+1$

$\sin (k+1)x=\sin kx\times \cos x+\cos kx\times \sin x$

Hence,
$|\sin (k+1)x|<|\sin \left(kx\right)|+|\sin x|$

since $|\cos x|\le 1$

Thus,

$|\sin (k+1)x|<k|\sin x|+|\sin x|<(k+1)|\sin x|$

Hence proved by Mathematical Induction.