Question

Find the expansion of cos(xsin(t)).

(a) \(\sum_{n=1}^∞ (\frac{x^n [Cos(nt)]}{n!})\)

(b) \(\sum_{n=0}^∞ (\frac{x^n [Cos(nt)]}{n!})\)

(c) \(\sum_{n=1}^∞ (\frac{x^n [Sin(nt)]}{n!})\)

(d) \(\sum_{n=0}^∞ (\frac{x^n [Sin(nt)]}{n!})\)

I got this question during a job interview.

Origin of the question is Taylor Mclaurin Series topic in division Differential Calculus of Engineering Mathematics

Answer 1

Right option is (b) \(\sum_{n=0}^∞ (\frac{x^n [Cos(nt)]}{n!})\)

The explanation is: Given, f(x)=Cos(xSin(t))=real part of (e^ixSin(t))

=real part of(e^xCos(t) e^ixSin(t))

=real part of(e^x[Cos(t)+iSin(t)])

=Real part of \(\sum_{n=0}^∞ \frac{x^n [Cos(nt)+iSin(nt)]}{n!}\)

=\(\sum_{n=0}^∞ \frac{x^n [Cos(nt)]}{n!}\)