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If sin^(-1)x + tan ^(-1) x = (pi)/(2), t...

If `sin^(-1)x + tan ^(-1) x = (pi)/(2)`, then prove that
`2x^(2) + 1 = sqrt(5)`

Text Solution

Verified by Experts

` sin^(-1) x + tan ^(-1) x =(pi)/(2)`
` rArr tan^(-1) x =(pi)/(2)- sin^(-1) x`
`" "=cos^(-1)x = tan^(-1).(sqrt(1-x^(2)))/(x)`
`rArr " "x=sqrt(1-x^(2))/(x)`
`rArr" " x^(2) = sqrt(1-x^(2))`
`" "x^(4)+x^(2) -1= 0`
`rArr " "x^(2)=(-1pmsqrt(1+4))/(2xx1)`
`rArr " "2x^(2) = -1 pm sqrt(5)`
`rArr " " 2x^(2) + 1 = sqrt(5)`
`" "(because 2x^(2)+ 1 = - sqrt(5)" is not possible")`
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