Question

The number of solutions of ${\sec }^2\theta +\cos e{c}^2\theta +2\cos e{c}^2\theta =8,0\le \theta \le \frac{\pi }{2}$ is

• A. 4
• B. 3
• C. 0
• D. 2

Answer 1

The correct option is D 2

$se{c}^2\theta +3cose{c}^2\theta =8\Rightarrow \frac{1}{co{s}^2\theta }+\frac{3}{si{n}^2\theta }=8$

$\Rightarrow si{n}^2\theta +3co{s}^2\theta =8si{n}^2\theta .co{s}^2\theta$

$\Rightarrow 1+2co{s}^2\theta =8co{s}^2\theta (1-co{s}^2\theta )$

$\Rightarrow 8co{s}^4\theta -6co{s}^2\theta +1=0$

$\Rightarrow (4co{s}^2\theta -1)(2co{s}^2\theta -1)=0$

but given $0\le \theta \le \frac{\pi }{2}$,therefore $cos\theta \ge 0$,

so

$cos\theta =\frac{1}{2},\frac{1}{\sqrt{2}}$

Hence solutions are,

$\theta =\frac{\pi }{4},\frac{\pi }{3}$

Hence, option 'D' is correct.