Question : If $\tan A=\frac{2}{5}$, then find the value of $\frac{\sec ^2 A}{\operatorname{cosec}^2 A}$.
Option 1: $\frac{2}{5}$
Option 2: $\frac{4}{25}$
Option 3: $\frac{9}{25}$
Option 4: $\frac{3}{5}$
Correct Answer: $\frac{4}{25}$
Solution : Given: $\tan A=\frac{2}{5}$ ⇒ $\cot A=\frac{5}{2}$ Now, $\sec^2 A = 1+ \tan^2 A$ ⇒ $\sec^2 A = 1+ (\frac{2}{5})^2$ ⇒ $\sec^2 A = 1+ (\frac{4}{25})$ ⇒ $\sec^2 A = (\frac{29}{25})$ Also, $\operatorname{cosec}^2 A = 1+ \cot^2 A$ ⇒ $\operatorname{cosec}^2 A = 1+ (\frac{5}{2})^2$ ⇒ $\operatorname{cosec}^2 A = 1+ (\frac{25}{4})$ ⇒ $\operatorname{cosec}^2 A = (\frac{29}{4})$ So, $\frac{\sec ^2 A}{\operatorname{cosec}^2 A} = \frac{\frac{29}{25}}{\frac{29}{4}}$ ⇒ $\frac{\sec ^2 A}{\operatorname{cosec}^2 A} = \frac{4}{25}$ Hence, the correct answer is $\frac{4}{25}$.
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