Question : If $\sec \theta=2 \frac{4}{23}$,find the value of $\tan^2 \theta$.
Option 1: $2 \frac{16}{529}$
Option 2: $1 \frac{200}{529}$
Option 3: $3 \frac{384}{529}$
Option 4: $3 \frac{177}{529}$
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Correct Answer: $3 \frac{384}{529}$
Solution :
Given:
$\sec \theta=2 \frac{4}{23}=\frac{50}{23}$
We know, $\tan^2\theta=\sec^2\theta-1$
⇒ $\tan^2\theta=(\frac{50}{23})^2-1$
⇒ $\tan^2\theta=\frac{2500}{529}-1$
⇒ $\tan^2\theta=\frac{1971}{529}$
$\therefore \tan^2\theta=3 \frac{384}{529}$
Hence, the correct answer is $3 \frac{384}{529}$.
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