Question : If $\sec\theta+\tan\theta=a$, then what is the value of $\sec\theta$?
Option 1: $\frac{a+1}{2a^2}$
Option 2: $\frac{(a-1)}{2a^2}$
Option 3: $\frac{a^2-1}{2a}$
Option 4: $\frac{a^2+1}{2a}$
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Correct Answer: $\frac{a^2+1}{2a}$
Solution :
Given: $\sec\theta+\tan\theta=a$ ---------------------(equation 1)
We know,
$\sec^2\theta–\tan^2\theta=1$
$⇒(\sec\theta+\tan\theta)(\sec\theta–\tan\theta)=1$
$⇒a(\sec\theta–\tan\theta)=1$
$⇒\sec\theta–\tan\theta=\frac{1}{a}$ -------------------(equation 2)
By solving equations 1 and 2 we get,
$\sec\theta=\frac{a+\frac{1}{a}}{2}$
$⇒\sec\theta=\frac{a^2+1}{2a}$
Hence, the correct answer is $\frac{a^2+1}{2a}$.
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