The number of real solutions of the equation sin - 1 ⁡ ∑ i = 1 ∞ x i + 1 - x ∑ i = 1 ∞ x 2 i = π 2 - cos - 1 ⁡ ∑ i = 1 ∞ - x 2 i - ∑ i = 1 ∞ - x i lying in the interval - 1 2 , 1 2 is____. (Here, the inverse trigonometric functions sin - 1 ⁡ x a n d cos - 1 ⁡ x assume values in - π 2 , π 2 a n d 0 , π respectively.)

Step-by-step solution

∑ i = 1 ∞ x i + 1 = x 2 1 - x ∑ i = 1 ∞ x 2 i = x 2 - x ∑ i = 1 ∞ - x 2 i = - x 2 + x ∑ i = 1 ∞ - x i = - x 1 + x For the given equation to have real solutions ∑ i = 1 ∞ x i + 1 - x ∑ i = 1 ∞ x 2 i = ∑ i = 1 ∞ - x 2 i - ∑ i = 1 ∞ - x i x 2 1 - x - x 2 2 - x = - x 2 + x + x 1 + x x x 3 + 2 x 2 + 5 x - 2 = 0 ⇒ x = 0   i s   a   s o l u t i o n , a l s o           f x = x 3 + 2