Question

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Answer 1

$\green{\large\underline{\sf{Given \:Question - }}}$

The number of solutions of the equation

$\rm :\longmapsto\:tan3x = cot4x, \: \: x \: \in \: [0, \: \pi]$

$\purple{\large\underline{\sf{Solution-}}}$

Given Trigonometric equation is

$\rm :\longmapsto\:tan3x = cot4x$

can be rewritten as

$\rm :\longmapsto\:\dfrac{sin3x}{cos3x} = \dfrac{cos4x}{sin4x}$

$\rm :\longmapsto\:cos4xcos3x = sin4xsin3x$

$\rm :\longmapsto\:cos4x \: cos3x - sin4x \: sin3x = 0$

We know,

$\red{ \boxed{ \sf{ \:cosxcosy - sinxsiny = cos(x + y)}}}$

So, using this identity, we get

$\rm :\longmapsto\:cos(4x + 3x) = 0$

$\rm :\longmapsto\:cos7x = 0$

We know,

$\red{ \boxed{ \sf{ \:cosx = 0\bf\implies \:x = (2n + 1)\dfrac{\pi}{2} \:\forall \: n \: \in \: Z}}}$

So, using this identity, we get

$\rm :\longmapsto\:7x = (2n + 1)\dfrac{\pi}{2}\:\forall \: n \: \in \: Z$

$\bf\implies \:\:x = (2n + 1)\dfrac{\pi}{14}\:\forall \: n \: \in \: Z$

As it is given that,

$\bf :\longmapsto\:x \: \in \: [0, \: \pi]$

So, x can assume the values, we have

$\green{\rm :\longmapsto\:x = \dfrac{\pi}{14}, \: \dfrac{3\pi}{14}, \: \dfrac{5\pi}{14}, \: \dfrac{7\pi}{14}, \: \dfrac{9\pi}{14}, \: \dfrac{11\pi}{14}, \: \dfrac{13\pi}{14}}$

Hence,

• Number of solutions = 7

Additional Information :-

$\purple{\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf T-eq & \bf Solution \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf sinx = 0 & \sf x = n\pi \:\forall \: n \: \in \: Z\\ \\ \sf cosx = 0 & \sf x = (2n + 1)\dfrac{\pi}{2}\:\forall \: n \: \in \: Z\\ \\ \sf tanx = 0 & \sf x = n\pi\:\forall \: n \: \in \: Z\\ \\ \sf sinx = siny & \sf x = n\pi + {( - 1)}^{n}y \:\forall \: n \: \in \: Z\\ \\ \sf cosx = cosy & \sf x = 2n\pi \pm \: y\:\forall \: n \: \in \: Z\\ \\ \sf tanx = tany & \sf x = n\pi + y \:\forall \: n \: \in \: Z\end{array}} \\ \end{gathered}\end{gathered}}$