The number of principal solutions of the equa | vedclass

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(D) The given equation is $	an 3x - 	an 2x - 	an x = 0$.Using the identity $	an(A-B) = \frac{	an A - 	an B}{1 + 	an A 	an B}$,we know that $

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more than $7$

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(D) The given equation is $	an 3x - 	an 2x - 	an x = 0$.Using the identity $	an(A-B) = \frac{	an A - 	an B}{1 + 	an A 	an B}$,we know that $	an 3x = 	an(2x + x) = \frac{	an 2x + 	an x}{1 - 	an 2x 	an x}$.This implies $	an 3x(1 - 	an 2x 	an x) = 	an 2x + 	an x$,which simplifies to $	an 3x - 	an 3x 	an 2x 	an x = 	an 2x + 	an x$.Rearranging gives $	an 3x - 	an 2x - 	an x = 	an 3x 	an 2x 	an x$.Thus,the original equation is equivalent to $	an 3x 	an 2x 	an x = 0$.This occurs when $	an 3x = 0$,$	an 2x = 0$,or $	an x = 0$.For $x \in [0, 2\pi)$,the solutions are:$1$. $	an x = 0 \implies x = 0, \pi$.$2$. $	an 2x = 0 \implies 2x = 0, \pi, 2\pi, 3\pi \implies x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$.$3$. $	an 3x = 0 \implies 3x = 0, \pi, 2\pi, 3\pi, 4\pi, 5\pi \implies x = 0, \frac{\pi}{3}, \frac{2\pi}{3}, \pi, \frac{4\pi}{3}, \frac{5\pi}{3}$.However,we must exclude values where $	an x, 	an 2x, 	ext{ or } 	an 3x$ are undefined.$	an x$ is undefined at $\frac{\pi}{2}, \frac{3\pi}{2}$.$	an 2x$ is undefined at $\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$.$	an 3x$ is undefined at $\frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{3\pi}{2}, \frac{11\pi}{6}$.Removing these,the valid solutions are $0, \pi, \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}$.There are $6$ distinct solutions,but the question asks for the number of principal solutions. Re-evaluating the set,we have $6$ values. Since $6$ is not an option and the set includes $0, \pi, \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}$,the count is $6$.

The number of principal solutions of the equation $\tan 3x - \tan 2x - \tan x = 0$ is:

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