If y-sin x-1sin x-sin 2x-1sin 2x-sin 3x-1sin nxsinn-1x- then dydx-

The correct option is B (n+1)^2(n+1)x ^2x Given, y=sinx[1sinx.sin2x+1sin2x.sin3x+...+1sinnx.sin(n+1)x] y=sinxsinx.sin2x+sinxsin2x.sin3x+...+sinxsi ...

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Byju's Answer

Standard XII

Mathematics

Formation of a Differential Equation from a General Solution

If y=sin x[...

Question

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 $\frac{d y}{d x}$ =

cot x - cot (n + 1) x

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(n + 1) {csc}^{2} (n + 1) x - {csc}^{2} x

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{csc}^{2} x - (n + 1) {csc}^{2} (n + 1) x

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cot x + cot (n + 1) x

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Solution

The correct option is B $(n + 1) {csc}^{2} (n + 1) x - {csc}^{2} x$
Given,
$y = s i n x [\frac{1}{s i n x . s i n 2 x} + \frac{1}{s i n 2 x . s i n 3 x} + . . . + \frac{1}{s i n n x . s i n (n + 1) x}]$
$y = \frac{s i n x}{s i n x . s i n 2 x} + \frac{s i n x}{s i n 2 x . s i n 3 x} + . . . + \frac{s i n x}{s i n n x . s i n (n + 1) x}$

$y = \frac{s i n (2 x - 3 x)}{s i n x . s i n 2 x} + \frac{s i n (3 x - 2 x)}{s i n 2 x . s i n 3 x} + . . . + \frac{s i n [(n + 1) x - n x]}{s i n n x . s i n (n + 1) x}$

$y = \frac{s i n 2 x . c o s x - c o s 2 x s i n x}{s i n x . s i n 2 x} + \frac{s i n 3 x . c o s 2 x - s i n 2 x c o s 3 x}{s i n 2 x . s i n 3 x} + . . . + \frac{s i n (n + 1) x c o s n x - c o s (n + 1) x s i n n x}{s i n n x . s i n (n + 1) x}$

$y = \frac{c o s x}{s i n x} - \frac{c o s 2 x}{s i n 2 x} + \frac{c o s 2 x}{s i n 2 x} - \frac{c o s x}{s i n 3 x} + . . . + \frac{c o s n x}{s i n n x} - \frac{c o s (n_{1}) x}{s i n (n + 1) x}$

$y = c o t x - c o t (n + 1) x$

$∴ \frac{d y}{d x} = - c o s e c^{2} x + c o s e c^{2} (n + 1) x \times (n + 1)$

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If y=sinx[1sinx.sin2x+1sin2x.sin3x++1sinnxsin(n+1)x] then dydx=

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 $\frac{d y}{d x}$ =