First slide

Introduction to ITF

Question

The values of $\sec^{2} (\tan^{- 1} 2) + {cosec}^{2} (\cot^{- 1} 3) =$

Moderate

A

5
B

10
C

15
D

20

Solution

We have,
$\begin{array}{l} \sec^{2} (\tan^{- 1} 2) + {cosec}^{2} (\cot^{- 1} 3) \\ = {\{\sec (\tan^{- 1} 2)\}}^{2} + {\{cosec (\cot^{- 1} 3)\}}^{2} \\ = {\{\sec (\tan^{- 1} \frac{2}{1})\}}^{2} + {\{cosec (\cot^{- 1} \frac{3}{1})\}}^{2} \\ = {\{\sec (\sec^{- 1} \sqrt{5})\}}^{2} + {\{cosec ({cosec}^{- 1} \sqrt{10})\}}^{2} \\ = (\sqrt{5})^{2} + (\sqrt{10})^{2} = 15 \end{array}$

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