Question

If ${\sin }^4\theta +\frac{1}{{\sin }^4\theta }=194,\theta \ne n\pi ,n\in Z$, then the value(s) of $(2\text{cosec}\theta -\cot \theta \cos \theta )$ can be

• A. $2$
• B. $-2$
• C. $4$
• D. $-4$

Answer 1

Answer: (C), (D)

The correct options are
C $4$
D $-4$

$2\text{cosec}\theta -\cot \theta \cos \theta =\frac{2-{\cos }^2\theta }{\sin \theta }=\frac{1+{\sin }^2\theta }{\sin \theta }=\sin \theta +\frac{1}{\sin \theta }$

Now,
${\sin }^4\theta +\frac{1}{{\sin }^4\theta }=194\Rightarrow {({\sin }^2\theta +\frac{1}{{\sin }^2\theta })}^2-2=194\Rightarrow {\sin }^2\theta +\frac{1}{{\sin }^2\theta }=14(\because {\sin }^2\theta >0)\Rightarrow {(\sin \theta +\frac{1}{\sin \theta })}^2-2=14\Rightarrow \sin \theta +\frac{1}{\sin \theta }=\pm 4\therefore 2\text{cosec}\theta -\cot \theta \cos \theta =\pm 4$