Question : If $\sec \theta+\tan \theta=5$, then $\operatorname{sin} \theta=$________.
Option 1: $0$
Option 2: $\frac{13}{12}$
Option 3: $\frac{12}{13}$
Option 4: $\frac{1}{5}$
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Correct Answer: $\frac{12}{13}$
Solution :
$\sec \theta+\tan \theta=5$---(1)
We know, $\sec^2 \theta-\tan^2 \theta=1$
⇒ $(\sec \theta+\tan \theta)(\sec \theta-\tan \theta)=5$
⇒ $\sec \theta-\tan \theta=\frac{1}{5}$---(2)
Solving equations 1 and 2, we get,
$\sec \theta=\frac{13}{5}$, $\tan\theta=\frac{12}{5}$
So, $\cos\theta=\frac{1}{\sec\theta}=\frac{5}{13}$
$\tan\theta=\frac{12}{5}$
⇒ $\frac{\sin\theta}{\cos\theta}=\frac{12}{5}$
⇒ $\frac{\sin\theta}{\frac{5}{13}}=\frac{12}{5}$
$\therefore \sin \theta=\frac{12}{13}$
Hence, the correct answer is $\frac{12}{13}$.
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