f(x) $= \frac{π}{4} + \frac{2}{π} [\frac{cos x}{1^{2}} + \frac{cos 3 x}{3^{2}} + . . . .] + [\frac{sin x}{1} + \frac{sin 2 x}{2} + \frac{sin 3 x}{3} + . . . .]$ The convergence of the above Fourier series at x = 0 gives

Q. If

1 + sin x + {sin}^{2} x + s i n^{3} x + . . . \infty

is equal to

4 + 2 \sqrt{3}, 0 \leq x < π

then x is equal to

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If 0≤x<π2, the the number of values of x for which sinx−sin2x+sin3x=0, is

The correct option is A 2sinx−sin2x+sin3x=0⇒(sinx+sin3x)−sin2x=0⇒2sin2x×cosx−sin2x=0⇒sin2x(2cosx−1)=0⇒sin2x=0 or cosx=12⇒x=0,π3

If $0 \leq x < \frac{π}{2}$ , the the number of values of $x$ for which $sin x - sin 2 x + sin 3 x = 0$ , is

The correct option is A $2$
$sin x - sin 2 x + sin 3 x = 0$

$\Rightarrow (sin x + sin 3 x) - sin 2 x = 0$

$\Rightarrow 2 sin 2 x \times cos x - sin 2 x = 0$

$\Rightarrow sin 2 x (2 cos x - 1) = 0$

$\Rightarrow sin 2 x = 0$ or $cos x = \frac{1}{2}$

$\Rightarrow x = 0, \frac{π}{3}$