Question : If $\sec\theta-\tan\theta=\frac{1}{\sqrt3}$, then the value of $\sec\theta.\tan\theta$ is:
Option 1: $\frac{2}{3}$
Option 2: $\frac{2}{\sqrt3}$
Option 3: $\frac{4}{\sqrt3}$
Option 4: $\frac{1}{\sqrt3}$
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Correct Answer: $\frac{2}{3}$
Solution :
Given: $\sec\theta-\tan\theta=\frac{1}{\sqrt3}$ -------------------------(1)
We know, $\sec^2\theta-\tan^2\theta=1$
⇒ $(\sec\theta+\tan\theta)(\sec\theta-\tan\theta)=1$
⇒ $(\sec\theta+\tan\theta)(\frac{1}{\sqrt3})=1$
⇒ $(\sec\theta+\tan\theta)=\sqrt3$ -----------------------------------(2)
Now solving equation (1) and equation (2) we get,
$\sec\theta=\frac{2}{\sqrt3}$ and $\tan\theta=\frac{1}{\sqrt3}$
So $\sec\theta.\tan\theta=\frac{2}{\sqrt3}×\frac{1}{\sqrt3}=\frac{2}{3}$
Hence, the correct answer is $\frac{2}{3}$.
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