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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