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If (tan^(-1) x)^2 + (cot^(-1) x)^2 = (5 ...

If `(tan^(-1) x)^2 + (cot^(-1) x)^2 = (5 pi^2) /8` find x.

Text Solution

Verified by Experts

The correct Answer is:
`x = -1`
`(tan^(-1) x)^(2) + (cot^(-1) x)^(2) = (5pi^(2))/(8)`
`rArr (tan^(-1)x + cot^(-1) x)^(2) 2 tan^(-1) x ((pi)/(2) - tan^(-1) x) = (5pi^(2))/(8)`
`rArr (pi^(2))/(4) - 2 xx (pi)/(2) tan^(-1) x + 2 (tan^(-1) x)^(2) = (5pi^(2))/(8)`
`rArr 2(tan^(-1) x)^(2) - pi tan^(-1) x - (3pi^(2))/(8) = 0`
`rArr tan^(-1) x = -(pi)/(4), (3pi)/(4)`
`rArr tan^(-1) = -(pi)/(4) " " ( :' tan^(-1) x in (-(pi)/(2), (pi)/(2)))`
`:. x = -1`
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