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Find the minimum value of `(sec^(-1) x)^(2) + (cosec^(-1) x)^(2)` |
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Answer» Let `I = (sec^(-1) x)^(2) + (cosec^(-1) x)^(2)` `= (sec^(-1) x + cosec^(-1) x)^(2) - 2 sec^(-1) x cosec^(-1) x` `= (pi^(2))/(4) - 2 sec^(-1) x ((pi)/(2) - sec^(-1) x)` `= (pi^(2))/(4) + 2 (sec^(-1) x)^(2) - pi sec^(-1) x` `= (pi^(2))/(4) + 2 [(sec^(-1) x)^(2) - 2(pi)/(4) sec^(-1) x + ((pi)/(4))^(2)] - (pi^(2))/(8)` `= 2 (sec^(-1) x - (pi)/(4))^(2) + (pi^(2))/(8) rArr I ge (pi^(2))/(8)` |
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