Question

Find the general solution of $\sec 4\theta -\sec 2\theta =2$.

• A. $\theta =\frac{2n\pi }{5}\pm \frac{\pi }{10}$
• B. $2n\pi \pm \frac{\pi }{5},n\in I$
• C. $2n\pi \pm \frac{\pi }{2},n\in I$
• D. $n\pi \pm \frac{\pi }{2},n\in I$

Answer 1

Answer: (A), (C)

The correct options are
A $\theta =\frac{2n\pi }{5}\pm \frac{\pi }{10}$
C $2n\pi \pm \frac{\pi }{2},n\in I$

$\sec 4\theta -\sec 2\theta =2$

$\Rightarrow \frac{1}{\cos 4\theta }-\frac{1}{\cos 2\theta }=2$

$\Rightarrow \cos 2\theta -\cos 4\theta =2\cos 2\theta \cos 4\theta$ for $\cos 2\theta \ne 0$ and $\cos 4\theta \ne 0$

$\Rightarrow \cos 2\theta -\cos 4\theta =\cos (2\theta +4\theta )+\cos (2\theta -4\theta )$

$\Rightarrow \cos 2\theta -\cos 4\theta =\cos 6\theta +\cos 2\theta$

$\Rightarrow \cos 6\theta +\cos 4\theta =0$

$\Rightarrow 2\cos \left(\frac{6\theta +4\theta }{2}\right)\cos \left(\frac{6\theta -4\theta }{2}\right)=0$

$\Rightarrow 2\cos 5\theta \cos \theta =0$

$\Rightarrow \cos 5\theta =0\Rightarrow 5\theta =2n\pi \pm \frac{\pi }{10}$

$\Rightarrow \theta =\frac{2n\pi }{5}\pm \frac{\pi }{10}$

or $\cos \theta =0\Rightarrow \theta =2n\pi +\frac{\pi }{2}$