Number of solutions of the equation tan^2 x + | vedclass

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(B) Given equation: $	an^2 x + 3 \cot^2 x = 2 \sec^2 x$Using the identity $\sec^2 x = 1 + 	an^2 x$,we get:$	an^2 x + 3 \cot^2 x = 2(1 + 	an^2 x)$$

Vedclass · Accepted Answer

$4$

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(B) Given equation: $	an^2 x + 3 \cot^2 x = 2 \sec^2 x$Using the identity $\sec^2 x = 1 + 	an^2 x$,we get:$	an^2 x + 3 \cot^2 x = 2(1 + 	an^2 x)$$	an^2 x + 3 \cot^2 x = 2 + 2 	an^2 x$$3 \cot^2 x - 	an^2 x = 2$Let $t = 	an^2 x$,then $\cot^2 x = \frac{1}{t}$.$3(\frac{1}{t}) - t = 2$$3 - t^2 = 2t$$t^2 + 2t - 3 = 0$$(t + 3)(t - 1) = 0$Since $t = 	an^2 x \ge 0$,we have $t = 1$.$	an^2 x = 1 \implies 	an x = \pm 1$.In the interval $[0, 2\pi]$,$	an x = 1$ at $x = \frac{\pi}{4}, \frac{5\pi}{4}$ and $	an x = -1$ at $x = \frac{3\pi}{4}, \frac{7\pi}{4}$.All these values are valid as $	an x$ and $\cot x$ are defined at these points.Thus,there are $4$ solutions.

Number of solutions of the equation $\tan^2 x + 3 \cot^2 x = 2 \sec^2 x$ lying in the interval $[0, 2\pi]$ is

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