Question : If $\sec \theta-\tan \theta=m$, then $2 \tan \theta=?$
Option 1: $\frac{1-m}{m} \\$
Option 2: $ \frac{1-m^2}{m} \\$
Option 3: $\frac{1+m^2}{m} \\$
Option 4: $ \frac{m^2-1}{m}$
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Correct Answer: $ \frac{1-m^2}{m} \\$
Solution : $\sec \theta-\tan \theta=m$ ---(i) $\sec^2 \theta-\tan ^2\theta=1$ $⇒(\sec \theta-\tan \theta)(\sec \theta+\tan \theta)=1$ $⇒(\sec \theta+\tan \theta)=\frac{1}{(\sec \theta-\tan \theta)}$ $⇒(\sec \theta+\tan \theta)=\frac{1}{m}$ ---(ii) Subtracting equation (ii) from (i), we get $⇒-2\tan \theta=m-\frac{1}{m}$ $⇒2\tan \theta=\frac{1}{m}-m$ $⇒2\tan\theta=\frac{1-m^2}{m}$ Hence, the correct answer is $\frac{1-m^2}{m}$.
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