Question

Find the value of x, if tan `[sec^(-1) (1/x) ] = sin ( tan^(-1) 2) , x > 0 `.

Answer 1

Let sec^-1 `(1/x) = theta`

` ⇒ sec theta = 1/x`

⇒ cos θ = x 

⇒ tan ` (sec^(-1) (1/x)) = tan theta = sqrt(1 -x^2 ) /x `                ...(1) 

Now consider, 

sin ( tan ^-1 2 )

Let tan^-1 2 = Φ

 tan Φ = 2 

sin ( tan^-1 2) = sin Φ = `2/sqrt(5) `            ...(ii) 

From (i) and (ii)

`sqrt(1- x^2 )/x = 2/sqrt(5)`

5(1 - x² ) = 4x² 

`x = +- sqrt(5)/3 " but " x > 0 ⇒ x = sqrt(5)/3`