Question

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

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

Answer 1

Answer: (C)

$tan\, \left(cot \,x\right) = cot\, \left(tan\, x\right) = tan \,\left(\frac{\pi}{2}-tan \,x\right)$
$\Rightarrow \quad cot\, x = n\, \pi +\frac{\pi }{2}-tan \,x$
$\left[\because tan \,\theta = tan\, \alpha \Rightarrow \theta = n\pi + \alpha\right]$
$\Rightarrow \quad cot \,x + tan\, x = n \,\pi +\frac{\pi }{2}$
$\Rightarrow \quad \frac{cos\, x}{sin\, x} + \frac{sin \,x}{cos\, x} = \left(2n+1\right)\frac{\pi }{2}$
$\Rightarrow \quad \frac{1}{sin \,x \,cos\, x} = \left(2n+1\right) \frac{\pi }{2}$
$\Rightarrow \quad \frac{1}{sin\, 2x} = \frac{\left(2n+1\right)\pi}{4}$
$\Rightarrow \quad sin \,2x = \frac{4}{\left(2n+1\right)\pi}$