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If Taylor series expansion of $\cos(x)$ is restricted to only first two terms and the permissible error is$ .54×10^{-2}$, then $x$ can be at most. Should i use the permissible error as this $\cos(x)-\left(1-\frac{x^{2}}{2!}\right)=.0054$

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What you need is $\displaystyle \left| \cos x - \left( 1 - \frac{x^2} 2 \right) \right| \le 0.0054$. So $\text{“}{\le}\text{''}$ rather than $\text{“}{=}\text{''}$, and you need the absolute value.

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