Question

find the general solution of tan2x=-cot(x+pi/2)​

Answer 1

Step-by-step explanation:

Tan2x = -cot[90+x]

tan2x = tanx

2x =x

x=0

Answer 2

$\large\underline{\sf{Given- }}$

$\rm :\longmapsto\:tan2x = - \: cot\bigg(x + \dfrac{\pi}{2} \bigg)$

$\large\underline{\sf{To\:Find - }}$

General solution

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

Given Trigonometric equation is

$\rm :\longmapsto\:tan2x = - \: cot\bigg(x + \dfrac{\pi}{2} \bigg)$

We know,

$\boxed{ \bf{ \: \: cot\bigg(x + \dfrac{\pi}{2} \bigg) = - \: tanx}}$

So, using this, we get

$\rm :\longmapsto\:tan2x = - ( - tanx)$

$\rm :\longmapsto\:tan2x = tanx$

$\boxed{ \bf{ \: tanx \: = \: tany \: \implies \: x = n\pi + y \: \forall \: n \in \: Z}}$

So, using this identity, we get

$\rm :\longmapsto\:2x = n\pi + x\: \: \: \: \: \forall \: n \in \: Z$

$\rm :\longmapsto\:2x - x = n\pi \: \: \: \: \: \forall \: n \in \: Z$

$\bf\implies \:\:x= n\pi \: \: \: \: \: \forall \: n \in \: Z$

Additional Information :-

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