The real solutions of the equation tan -1 x x + 1 + sin -1 x 2 + x + 1 = π 2 are

Question

The real solutions of the equation tan -1 x x + 1 + sin -1 x 2 + x + 1 = π 2 are, - 1 , 0, 0 , 1, - 1 , 1, - 1 , 2

MARKS App · Accepted Answer

tan -1 x x + 1 + sin -1 x 2 + x + 1 = π 2 ⇒ tan -1 x 2 + x + sin -1 x 2 + x + 1 = π 2 This equation holds true, if x 2 + x ≥ 0 and 0 ≤ x 2 + x + 1 ≤ 1 Now, x 2 + x ≥ 0 and 0 ≤ x 2 + x + 1 ≤ 1 ⇒ x 2 + x ≥ 0 and x 2 + x + 1 ≤ 1 ∵ x 2 + x + 1 > 0 for all x ⇒ x 2 + x ≥ 0 and x 2 + x ≤ 0 ⇒ x 2 + x = 0 ⇒ x = 0 , - 1 . Clearly, these two values satisfy the given equation. Hence, x = - 1 , 0 are the solutions of the given equation.

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