This article is devoted to obtain some sharp lower and upper bounds for $exp(x^2)$ in the interval $(− π /2 , π/ 2)$. The bounds are of the type $[(a+f (x))/(a+1)]^b$, where $f (x)$ is cosine or hyperbolic cosine. The results are then used to obtain and refine some known Cusa-Huygens type inequalities. In particular, new simple proof of Cusa-Huygens type inequalities is presented as an application. For other interesting applications of the main results, sharp bounds of the truncated Gaussian sine integral and error function are established. They can be useful in probability theory.