Question : If $\sec\theta+\tan\theta=5$, then find the value of $\tan\theta$.
Option 1: $\frac{5}{12}$
Option 2: $\frac{13}{5}$
Option 3: $\frac{13}{3}$
Option 4: $\frac{12}{5}$
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Correct Answer: $\frac{12}{5}$
Solution :
Given: $\sec\theta+\tan\theta=5$--------------------(equation 1)
We know,
$\sec^2\theta-\tan^2\theta=1$
⇒ $(\sec\theta+\tan\theta)(\sec\theta-\tan\theta)=1$
⇒ $(5)(\sec\theta-\tan\theta)=1$
⇒ $(\sec\theta-\tan\theta)=\frac{1}{5}$----------------------(equation 2)
Now, solving equations 1 and 2 we get,
$\tan\theta=\frac{12}{5}$
Hence, the correct answer is $\frac{12}{5}$.
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