A) \[\sum\limits_{k=1}^{n}{k\,\tan \,k\,x}\]
B) \[y.\sum\limits_{k=1}^{n}{k\,\cot \,kx}\]
C) \[y.\sum\limits_{k=1}^{n}{k\,\tan \,kn}\]
D) \[\sum\limits_{k=1}^{n}{\cot kx}\]
Correct Answer: B
Solution :
Given, \[y=\sin x.\sin 2x.sin3x.....sin\,nx.\] Taking log on both sides, \[\log \,\,y=\log \,\sin x+\log \,\sin 2x+....+\log \,\sin \,nx\] Differentiating w.r.t. x, \[\frac{1}{y}.\frac{dy}{dx}=1.\cot x+2\cot 2x+....+n\cot nx\] \[\Rightarrow \] \[\frac{dy}{dx}=y.\sum\limits_{k=1}^{n}{k\,\cot \,kx}\] \[\Rightarrow \] \[y'=y'\,\sum\limits_{k=1}^{n}{k\,\cot \,kx}\]
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