Question

Find the real solutions of the equation
`tan^-1 sqrt(x(x + 1)) + sin^-1 sqrt(x^2 + x + 1) = pi/2`

Answer 1

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