Number of Real Roots of Trigonometric Exponential Equation

By AM-GM inequality, $2^x + 2^{-x} \ge 2\sqrt{2^x \cdot 2^{-x}} = 2$. Also, $2\cos(x(x+1)) \le 2$. Equality holds if and only if both sides equal 2. RHS $= 2 \implies 2^x = 1 \implies x = 0$. Checking LHS for $x = 0$: $2\cos(0) = 2$, which satisfies the equation. So $x = 0$ is the only real root.