Question

If $\cot\theta + \tan\theta = m$and $\sec\theta - \cos\theta = n,$ then which of the following is correct

• A. $m(mn^{2})^{1/3} - n(nm^{2})^{1/3} = 1$
• B. $m(m^{2}n)^{1/3} - n(mn^{2})^{1/3} = 1$
• C. $n(mn^{2})^{1/3} - m(nm^{2})^{1/3} = 1$
• D. $n(m^{2}n)^{1/3} - m(mn^{2})^{1/3} = 1$

Answer 1

Answer: (A)

$m(mn^{2})^{1/3} - n(nm^{2})^{1/3} = 1$

Answer 2

As given $\frac{1}{\tan\theta} + \tan\theta = m \Rightarrow 1 + \tan^{2}\theta = m\tan\theta$

$\Rightarrow \sec^{2}\theta = m\tan\theta$ …..(i)

and $\sec\theta - \cos\theta = n \Rightarrow \sec^{2}\theta - 1 = n\sec\theta$

$\Rightarrow \tan^{2}\theta = n\sec\theta$

$\Rightarrow \tan^{4}\theta = n^{2}\sec^{2}\theta = n^{2}.m\tan\theta$ {by (i)}

$\Rightarrow \tan^{3}\theta = n^{2}m,(\because\tan\theta \neq 0)$

$\Rightarrow \tan\theta = (n^{2}m)^{1/3}$ …..(ii)

Also, $\sec^{2}\theta = m\tan\theta = m(n^{2}m)^{1/3}$ {by (i) and (ii)}

∴ Using the identity $\sec^{2}\theta - \tan^{2}\theta = 1$

$\Rightarrow m(mn^{2})^{1/3} - (n^{2}m)^{2/3} = 1$

$\Rightarrow m(mn^{2})^{1/3} - n(nm^{2})^{1/3} = 1.$