Question

If ${\tan }^3\theta +{\cot }^3\theta =2,$ then which of the following is/are correct?

• A. $\sin \theta =-\sec \theta$
• B. $\sin \theta =-\cos \theta$
• C. $2\sin \theta =\sec \theta$
• D. $2\sin \theta =\cos \theta$

Answer 1

The correct option is C $2\sin \theta =\sec \theta$

${\tan }^3\theta +{\cot }^3\theta =2$
Assuming ${\tan }^3\theta =t$, so
$t+\frac{1}{t}=2\Rightarrow t^2-2t+1=0\Rightarrow (t-1{)}^2=0\Rightarrow t=1\Rightarrow {\tan }^3\theta =1\therefore \tan \theta =1$
Now, $\sin \theta =\frac{1}{\sqrt{2}}$
and $\cos \theta =\frac{1}{\sqrt{2}}$
Therefore, $2\sin \theta =\sec \theta$