displaystyle∑r=0n n C r sin r x cos (n-r) x=

Question

displaystyle∑r=0n n C r sin r x cos (n-r) x= (A) 2 n -1 sin (n-1) x (B) 2 n sin n x (C) 2 n -1 sin n x (D) none of these

Tardigrade Admin · Accepted Answer

We have, displaystyle∑r=0n Cr sin r x cos (n-r) x = (1/2)[(n C0 sin 0 x cos n x+n C n sin n x cos 0 x). +(n C1 sin x cos (n-1) x+n C n-1 sin (n-1) x ⋅ cos x) +(n C2 sin 2 x cos (n-2) x+n C n-2 sin (n-2) x ⋅ cos 2 x) .+ ldots+(n C n sin n x cos 0 x+n C0 sin 0 x cos n x)] = (1/2)[n C0 sin n x+n C1 sin n x+ ldots+n C n sin n x] = (1/2)[n C0+n C1+ ldots+n C n] sin n x=(2n sin n x/2) ∴ displaystyle∑r=0n Cr sin r x cos (n-r) x=2 n-1 sin n x.

Q. $r = 0 \sum n^{n} C_{r} sin r x cos (n - r) x =$

$2^{n - 1} sin (n - 1) x$

$2^{n} sin n x$

$2^{n - 1} sin n x$

none of these

Q. r=0∑n​nCr​sinrxcos(n−r)x=

2n−1sin(n−1)x

2nsinnx

2n−1sinnx

none of these

Q. $r = 0 \sum n^{n} C_{r} sin r x cos (n - r) x =$

$2^{n - 1} sin (n - 1) x$

$2^{n} sin n x$

$2^{n - 1} sin n x$