Question

If ${\tan }^2\theta -\sec \theta =5$, then $\cos \theta =\frac{1}{3}$.

• A. True
• B. False

Answer 1

The correct option is A True

${\tan }^2\theta -\sec \theta =5$

$\Rightarrow {\sec }^2\theta -1-\sec \theta =5$

$\Rightarrow {\sec }^2\theta -\sec \theta -6=0$

$\Rightarrow {\sec }^2\theta +2\sec \theta -3\sec \theta -6=0$

$\Rightarrow \sec \theta (\sec \theta +2)-3(\sec \theta +2)=0$

$\Rightarrow (\sec \theta +2)(\sec \theta -3)=0$

$\Rightarrow (\sec \theta +2)=0$ or $(\sec \theta -3)=0$

$\Rightarrow \sec \theta =-2$ or $\sec \theta =3$

$\Rightarrow \frac{1}{\cos \theta }=-2$ or $\frac{1}{\cos \theta }=3$

$\Rightarrow \cos \theta =\frac{-1}{2}$ or $\cos \theta =\frac{1}{3}\because \cos \theta \in [-1,1]$

$\therefore \cos \theta =\frac{1}{3}$