This article is devoted to the determination of sharp lower and upper bounds for exp(−x^2) over the interval (−epsilon, epsilon). The bounds are of the type [a+f(x)/ a+1]^α, where f(x) denotes either cosine or hyperbolic cosine. The results are then used to obtain and refine some known Cusa-Huygens type inequalities. In particular, a 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 functions are established. They can be useful in probability theory.