Question : If $\sec \ A + \tan \ A = 3$, then $\cos \ A$ is equal to:
Option 1: $\frac{4}{3}$
Option 2: $\frac{3}{5}$
Option 3: $\frac{3}{4}$
Option 4: $\frac{5}{3}$
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Correct Answer: $\frac{3}{5}$
Solution :
Given: $\sec\ A + \tan \ A = 3$ ---(1)
We know, $\sec^2A-\tan^2A =1$
⇒ $(\sec\ A - \tan \ A)(\sec \ A + \tan\ A) = 1$
⇒ $(\sec\ A - \tan\ A) = \frac{1}{3}$ ---(2)
On adding the equations 1 and 2, we get,
$(\sec \ A + \tan \ A) + (sec \ A - \tan \ A) = 3+\frac{1}{3}$
⇒ $2\sec\ A = \frac{10}{3}$
$\therefore \sec \ A = \frac{5}{3}$
The value of $\cos \ A = \frac{1}{\sec \ A}=\frac{3}{5}$
Hence, the correct answer is $\frac{3}{5}$.
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