If \(\sec\theta-\tan\theta=\displaystyle\frac{1}{\sqrt{3}}\), then what is the value of \(\sec\theta\)?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{2}{\sqrt{3}}\)
The identity \(\sec^2\theta-\tan^2\theta=1\) factorises as \((\sec\theta+\tan\theta)(\sec\theta-\tan\theta)=1\). So the given difference also fixes the sum, and adding the two gives \(2\sec\theta\).
\(\sec\theta+\tan\theta=\displaystyle\frac{1}{1/\sqrt{3}}=\sqrt{3}\)
Adding: \(2\sec\theta=\sqrt{3}+\displaystyle\frac{1}{\sqrt{3}}=\frac{3+1}{\sqrt{3}}=\frac{4}{\sqrt{3}}\)
\(\sec\theta=\displaystyle\frac{2}{\sqrt{3}}\)