I am assuming you do not have the observed data, and just have the means and standard deviations. Let the data from the left axilla be $X_1, X_2, \dots, X_n$, and let the data from the right axilla be $Y_1, Y_2, \dots, Y_n$. I am assuming that you have equal sample sizes.
(Before I go on, I must warn you that your pooled/combined sample is no longer independent, because you have two data points on each individual. This is important if you are going to use the standard deviations to test a hypothesis.)
You are given the quantities $\bar{X}_n = 36.4$ and $sd(X_1) = 0.2$ and $\bar{Y}_n = 36.7$ and $sd(Y_1) = .18$.
(Here $\bar{X}_n$ denotes sample mean of $X$s similarly for $Y$).
If you were to pool all the data, you would have $X_1, \dots, X_n, Y_1, \dots, Y_n$. Let these be noted by $Z_1, \dots, Z_{2n}$. Then mean of $Z$,
$$\bar{Z}_{2n} =\dfrac{X_1 + X_2 + \dots + X_n + Y_1 +Y_2 + \dots Y_n}{2n} = \dfrac{\bar{X}_n + \bar{Y}_n}{2} = 36.55$$
The standard deviation can be calculated using pooled estimators.
$$sd(Z_1)^2 = \dfrac{(n-1)sd(X_1)^2 + (n-1)sd(Y_1)^2}{2(n-1)} = \dfrac{sd(X_1)^2 + sd(Y_1)^2}{2}.$$
Thus the standard deviation is
$$sd(Z_1) = \sqrt{\dfrac{sd(X_1)^2 + sd(Y_1)^2}{2}}. $$
Like I mentioned before, since your sample is not independent, note that
$$sd(\bar{Z}_{2n})^2 \ne \dfrac{sd(Z_1)^2}{2n}$$.