Question

If tan (cot x) = cot (tan x), then

• A. $sin2x=\frac{2}{\left(2n+1\right)\pi }$
• B. $sinx=\frac{4}{\left(2n+1\right)\pi }$
• C. $sin2x=\frac{4}{\left(2n+1\right)\pi }$
• D. None of these

Answer 1

Answer: (C)

$tan\left(cotx\right)=cot\left(tanx\right)=tan\left(\frac{\pi }{2}-tanx\right)$
$\Rightarrow cotx=n\pi +\frac{\pi }{2}-tanx$
$\left[\because tan\theta =tan\alpha \Rightarrow \theta =n\pi +\alpha \right]$
$\Rightarrow cotx+tanx=n\pi +\frac{\pi }{2}$
$\Rightarrow \frac{cosx}{sinx}+\frac{sinx}{cosx}=\left(2n+1\right)\frac{\pi }{2}$
$\Rightarrow \frac{1}{sinxcosx}=\left(2n+1\right)\frac{\pi }{2}$
$\Rightarrow \frac{1}{sin2x}=\frac{\left(2n+1\right)\pi }{4}$
$\Rightarrow sin2x=\frac{4}{\left(2n+1\right)\pi }$