If y=sin x1sin x sin 2x+1sin 2 x sin 3x+...+1sin n x sin(n+1)x then dydx=

A
$\cot x - \cot (n + 1) x$
B
$(n + 1) {cosec}^{2} (n + 1) x - {cosec}^{2} x$
C
${cosec}^{2} x - (n + 1) {cosec}^{2} (n + 1) x$
D
$\cot x + \cot (n + 1) x$

Solution:

The given equaiton is

$\begin{array}{rcl} y & = & \frac{\sin x}{\sin x \sin 2 x} + \frac{\sin x}{\sin 3 x si n 2 x} + . . . + \frac{\sin x}{\sin (n + 1) x \sin n x} \\ = & \frac{\sin (2 x - x)}{\sin x \sin 2 x} + \frac{\sin (3 x - 2 x)}{\sin 3 x \sin 2 x} + . . . + \frac{\sin ((n + 1) x - n x)}{\sin (n + 1) x \sin n x} \\ = & c o t x - c o t 2 x + c o t 2 x - c o t 3 x + . . . + c o t n x - c o t (n + 1) x \\ = & c o t x - c o t (n + 1) x \end{array}$

Differentiate both sides

$\frac{d y}{d x} = - \cos e c^{2} x + (n + 1) \cos e c^{2} (n + 1) x$

If $y = \sin x (\frac{1}{\sin x \sin 2 x} + \frac{1}{\sin 2 x \sin 3 x} + . . . + \frac{1}{\sin n x \sin (n + 1) x})$ then $\frac{dy}{dx} =$

Solution:

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