Question

If $X=n\pi -{\tan }^{-1}3$ is a solution of the equation $12\tan 2x+\frac{\sqrt{10}}{\cos x}+1=0$ if

• A. $n$ is an any integer
• B. $n$ is an odd integer
• C. $n$ is a positive integer
• D. $n\in 2M,M\in l$

Answer 1

Answer: (B)

$n$ is an odd integer

$12\tan 2x+\frac{\sqrt{10}}{\cos x}+1=0$
$\Rightarrow 12\tan (2n\pi -2{\tan }^{-1}3)+\frac{\sqrt{10}}{\cos (n\pi -{\tan }^{-1}3)}+1=0$
Case 1) if $n$ is even
$-12\tan \left(2{\tan }^{-1}3\right)+\frac{\sqrt{10}}{\cos \left({\tan }^{-1}3\right)}+1=0$
$\Rightarrow -12\left(\frac{-3}{4}\right)+\frac{\sqrt{10}}{\left(\frac{1}{\sqrt{10}}\right)}+1=0$
$\Rightarrow +9+10+1=0\Rightarrow 20\ne 0$
Case 2) If $n$ is odd
$-12\tan \left(2{\tan }^{-1}3\right)+\frac{\sqrt{10}}{-\cos \left({\tan }^{-1}3\right)}+1=0$
$\Rightarrow -12\left(\frac{-3}{4}\right)-\frac{\sqrt{10}}{\left(\frac{1}{\sqrt{10}}\right)}+1=0$
$\Rightarrow +9-10+1=0$