Question : If cos A = $\frac{5}{13}$, then what is the value of (tan A + cot A)?
Option 1: $\frac{60}{169}$
Option 2: $\frac{109}{169}$
Option 3: $\frac{169}{60}$
Option 4: $\frac{169}{109}$
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Correct Answer: $\frac{169}{60}$
Solution :
Given that $\cos A = \frac{5}{13}$,
So, $\sin A = \sqrt{1 - \cos^2 A} = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \frac{12}{13}$
$⇒\tan A = \frac{\sin A}{\cos A}$ and $\cot A = \frac{1}{\tan A}$
$⇒\tan A = \frac{\sin A}{\cos A} = \frac{12}{5}$
$⇒\cot A = \frac{1}{\tan A} = \frac{5}{12}$
$\therefore \tan A + \cot A = \frac{12}{5} + \frac{5}{12} = \frac{144 + 25}{60} = \frac{169}{60}$
Hence, the correct answer is $\frac{169}{60}$.
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