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