Question

If tan 4A.tan 6A = 1, Find the value of A.​

Answer 1

Step-by-step explanation:

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Answer 2

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

Given that,

$\rm :\longmapsto\:tan4A \: tan6A = 1$

We know that,

$\green{ \boxed{ \sf \: tanx = \dfrac{sinx}{cosx}}}$

Using this, we get

$\rm :\longmapsto\:\dfrac{sin4A}{cos4A} \times \dfrac{sin6A}{cos6A} = 1 \:$

$\rm :\longmapsto\:\dfrac{sin4A \: sin6A}{cos4A \: cos6A} = 1$

$\rm :\longmapsto\:cos6A \: cos4A = sin6A \: sin4A$

$\rm :\longmapsto\:cos6A \: cos4A - sin6A \: sin4A = 0$

$\green{ \boxed{ \because\bf \: cosxcosy - sinxsiny = cos(x + y)}}$

$\rm :\longmapsto\:cos(6A + 4A) = 0$

$\rm :\longmapsto\:cos(10A) = 0$

$\rm :\longmapsto\:10A = \bigg(2n + 1 \bigg)\dfrac{\pi}{2} \: where \: n \: \in \: Z$

$\boxed{ \because \sf{ \: cosx = 0 \implies \: x = \bigg(2n + 1 \bigg)\dfrac{\pi}{2} \: where \: n \: \in \: Z}}$

$\rm :\longmapsto\:A = \bigg(2n + 1 \bigg)\dfrac{\pi}{20} \: where \: n \: \in \: Z$

Additional Information :-

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

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \purple{\boxed{ \bf \: where \: n \: \in \: \: Z}}$