Question : If $\cos \theta=\frac{5}{13}$, then the value of $\tan ^2 \theta+\sec ^2 \theta$ is equal to:
Option 1: $\frac{303}{25}$
Option 2: $\frac{313}{25}$
Option 3: $\frac{233}{25}$
Option 4: $\frac{323}{25}$
Correct Answer: $\frac{313}{25}$
Solution :
Given: The value of $\cos \theta=\frac{5}{13}$.
Use the trigonometric identity, $\sin^2 \theta +cos^2 \theta=1$.
$\tan ^2 \theta+\sec ^2 \theta=\frac{\sin^2 \theta}{\cos^2 \theta}+\frac{1}{\cos^2 \theta}$
= $\frac{\sin^2 \theta+1}{\cos^2 \theta}$
= $\frac{1–cos^2 \theta+1}{\cos^2 \theta}$
= $\frac{2–cos^2 \theta}{\cos^2 \theta}$
= $\frac{2–(\frac{5}{13})^2}{(\frac{5}{13})^2}$
= $\frac{2–\frac{25}{169}}{\frac{25}{169}}$
= $\frac{(338–25)\times 169}{169\times 25}$
= $\frac{313}{25}$
Hence, the correct answer is $\frac{313}{25}$.
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