For any real value of x, the expression 2cos(x(x+1))=2x+2−x would always be positive. Lets find the maximum value of 2cos(x(x+1))=2x+2−x. Applying AM-GM inequality we have
2x+2−x
2
≥√2x×2−x ⇒2x+2−x≥2√20 ⇒2x+2−x≥2 Therefore, 2cos(x(x+1))≥2 It is known that −1≤cosθ≤1 ⇒2cos(x(x+1))=2 Hence, the expression is valid only if 2x+2−x=2, which is true for only one value of x i.e. 0. Therefore, the expression has only one real solution.