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Question
Find the real solutions of the equation
`tan^-1 sqrt(x(x + 1)) + sin^-1 sqrt(x^2 + x + 1) = pi/2`
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Solution
We have `tan^-1 sqrt(x(x + 1)) + sin^-1 sqrt(x^2 + x + 1) = pi/2`
⇒ `tan^-1 sqrt(x(x +1)) = pi/2 - sin^-1 sqrt(x^2 + x + 1)`
= `cos^-1 sqrt(x^2 + x + 1)`
= `tan^-1 sqrt(-x^2 - x)/sqrt(x^2 +x + 1)` ....(From the figure)
⇒ `sqrt(x(x + 1)) = sqrt(-x^2 - x)/sqrt(x^2 + x + 1)`
⇒ `x^2 + x` = 0
⇒ x = 0, –1
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