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Trigonometrical Equations
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The solution set of the equation $tan(\pi\, tanx) = cot(\pi\, cot\, x)$ is
A
$\phi $
B
$\{0\}$
C
$\left\{ {\frac{\pi }{4}} \right\}$
D
none of these
Solution
$\tan (\pi \tan x)=\cot (\pi \cot x)$
$=\tan \left(\frac{\pi}{2}-\pi \cot x\right)$
$\Rightarrow \quad \pi \tan x=\frac{\pi}{2}-\pi \cot x$
$\Rightarrow \quad \tan x+\cot x=\frac{1}{2}$
$\Rightarrow \quad \tan x+\frac{1}{\tan x}=\frac{1}{2}$
$\Rightarrow \quad 2 \tan ^{2} x+2=\tan x$
$\Rightarrow \quad 2 \tan ^{2} x-\tan x+2=0$
This is quadratic in $\tan \mathrm{x}$.
$\Rightarrow \quad \tan x=\frac{1 \pm \sqrt{1-4.4}}{4}=\frac{1 \pm \sqrt{-15}}{2}$
which are not real
$\therefore $ solution set $=\phi$
Standard 11
Mathematics
