First slide

Properties of ITF

Question

The value of $cosec (2 \cot^{- 1} 5 + \cos^{- 1} \frac{4}{5})$ is

Moderate

A

$\frac{65}{56}$
B

$\frac{56}{65}$
C

$\frac{63}{65}$
D

$\frac{65}{63}$

Solution

$cosec (2 \cot^{- 1} 5 + \cos^{- 1} \frac{4}{5})$
$= cosec (2 \tan^{- 1} \frac{1}{5} + \tan^{- 1} \frac{3}{4})$
$\begin{array}{l} = cosec (\tan^{- 1} (\frac{2 \times \frac{1}{5}}{1 - \frac{1}{25}}) + \tan^{- 1} \frac{3}{4}) \\ = cosec (\tan^{- 1} \frac{5}{12} + \tan^{- 1} \frac{3}{4}) = cosec \{\tan^{- 1} (\frac{\frac{5}{12} + \frac{3}{4}}{1 - \frac{5}{12} \times \frac{3}{4}})\} \\ = cosec (\tan^{- 1} \frac{56}{33}) = cosec ({cosec}^{- 1} \frac{65}{56}) = \frac{65}{56} \end{array}$

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