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Prove that `tan^(-1) {(x)/(a + sqrt(a^(2) - x^(2)))} = (1)/(2) sin^(-1).(x)/(a), -a lt x lt a` |
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Answer» Let `x = a sin theta, -a lt x lt a`. Then, `-a lt a sin theta lt a` or `-1 lt sin theta lt 1 rArr theta in (-(pi)/(2), (pi)/(2))` `rArr tan^(-1) {(x)/(a+ sqrt(a^(2) - x^(2)))} = tan^(-1) {(a sin theta)/(a + sqrt(a^(2) - a^(2) sin^(2) theta))}` `= tan^(-1) {(sin theta)/(1 + cos theta)}` `= tan^(-1) {(2 sin (theta//2) cos (theta//2))/(2 cos^(2) (theta//2))}` `= tan^(-1) {tan.(theta)/(2)}` `= (theta)/(2) = (1)/(2) sin^(-1). (x)/(a)` |
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