First slide

Properties of ITF

Question

If ${(T a n^{- 1} x)}^{2} + {(C o t^{- 1} x)}^{2} = \frac{5 π^{2}}{8}$ then $x =$

Difficult

A

-1
B

1
C

0
D

2

Solution

${(T a n^{- 1} x)}^{2} + {(C o t^{- 1} x)}^{2} = \frac{5 π^{2}}{8} \Rightarrow {(T a n^{- 1} x + C o t^{- 1} x)}^{2} - 2 T a n^{- 1} x C o t^{- 1} x = \frac{5 π^{2}}{8}$
$\begin{array}{l} \Rightarrow {({Tan}^{- 1} x + {Cot}^{- 1} x)}^{2} - 2 {Tan}^{- 1} x (\frac{π}{2} - {Tan}^{- 1} x) = \frac{5 π^{2}}{8} \\ \Rightarrow 2 {({Tan}^{- 1} x)}^{2} - π {Tan}^{- 1} x - \frac{3 π^{2}}{8} = 0 \\ \Rightarrow {Tan}^{- 1} x = \frac{π}{4}, \frac{3 π}{4} \\ B u t T a n^{- 1} x = \frac{π}{4} c a n n o t s a t i s f y t h e e q u a t i o n . \\ \Rightarrow T a n^{- 1} x = \frac{3 π}{4} \\ \Rightarrow x = - 1 \end{array}$

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