Question : If $0\leq \theta\leq \frac{\pi}{2}$ and $\sec^{2}\theta+\tan^{2}\theta=7$, then $\theta$ is:
Option 1: $\frac{5\pi}{12}$
Option 2: $\frac{\pi}{3}$
Option 3: $\frac{\pi}{5}$
Option 4: $\frac{\pi}{6}$
Correct Answer: $\frac{\pi}{3}$
Solution :
Given: $\sec^{2}\theta+\tan^{2}\theta=7$ -------------(1)
We know that $\sec^{2}\theta-\tan^{2}\theta=1$ ------------(2)
Adding equations (1) and (2), we have,
⇒ $2\sec^{2}\theta=8$
⇒ $\sec^{2}\theta=4$
⇒ $\sec\theta=2$
⇒ $\sec\theta=\sec60°$
⇒ $\theta=60°=\frac{\pi}{3}$
Hence, the correct answer is $\frac{\pi}{3}$.
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