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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