InterviewSolution
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InterviewSolution
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If `(1)/(sqrt2) lt x lt 1`, then prove that `cos^(-1) x + cos^(-1) ((x + sqrt(1 - x^(2)))/(sqrt2)) = (pi)/(4)` |
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Answer» `cos^(-1) x + cos^(-1) ((x + sqrt(1 - x^(2)))/(sqrt2))` Let `cos^(-1) x = theta " or " x = cos theta` For `(1)/(sqrt2) lt x lt 1, 0 lt theta lt (pi)/(4)` `rArr cos^(-1) x + cos^(-1) (x + sqrt(1 - x^(2))/(sqrt2))` `= theta + cos^(-1) ((cos theta + sqrt(1 - cos^(2) theta))/(sqrt2))` `= theta + cos^(-1) ((cos theta + sin theta)/(sqrt2))` `= theta + cos^(-1) (cos (theta - (pi)/(4)))` Now, `(theta - (pi)/(4)) in (-(pi)/(4), 0)`, which is not the principal values of `cos^(-1)` function But `((pi)/(4) - theta) in (0, (pi)/(4))` `rArr theta + cos^(-1) (cos (theta - (pi)/(4))) = theta + cos^(-1) (cos ((pi)/(4) - theta))` `= theta + ((pi)/(4) - theta)` `= (pi)/(4)` |
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