Question : If $\sec \theta+\tan \theta=5, (\theta \neq 0)$, then $\sec \theta$ is equal to:
Option 1: $\left(5+\frac{1}{5}\right)$
Option 2: $\frac{1}{2}\left(3+\frac{1}{3}\right)$
Option 3: $\frac{1}{2}\left(5+\frac{1}{5}\right)$
Option 4: $\left(3+\frac{1}{3}\right)$
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Correct Answer: $\frac{1}{2}\left(5+\frac{1}{5}\right)$
Solution :
Given: $\sec \theta+\tan \theta=5$ ..... equation1
$\sec^{2}\theta - \tan^{2}\theta =1$
⇒ $(\sec \theta+\tan \theta)(\sec \theta-\tan \theta)=1$
⇒ $\sec \theta-\tan \theta=\frac{1}{5}$......equation2
Adding equation1 and equation2
$2\sec\theta = 5 + \frac{1}{5}$
⇒ $\sec\theta=\frac{1}{2}\left(5+\frac{1}{5}\right)$
Hence, the correct answer is $\frac{1}{2}\left(5+\frac{1}{5}\right)$.
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