Integer Trigonometry!

Since $m(x)$ is a continuous, differentiable (and periodic) function for all $x \in \mathbb{R}$ it suffices to find the minimum and maximum and then count the number of integer values in between (and inclusive).

Now we can rewrite the function as

$m(x) = 144^{1 - \cos^{2}(x)} + 144^{\cos^{2}(x)} = \dfrac{144}{144^{\cos^{2}(x)}} + 144^{\cos^{2}(x)} = \dfrac{144}{y} + y,$

where $y = 144^{\cos^{2}(x)}$ ranges from $1$ to $144$ since $0 \le \cos^{2}(x) \le 1.$

Next, with $m$ now expressed as $m(y) = \dfrac{144}{y} + y$ we have that

$m'(y) = -\dfrac{144}{y^{2}} + 1 = 0$ when $y = 12,$

(ignoring the negative root since $y \ge 1$ ). Now $m''(y) = \frac{288}{y^{3}} \gt 0$ for $y = 12,$ (and in fact for the entire range of $y$ ), so by the second derivative test we know that the minimum is $m(12) = 24.$ Since there are no other positive critical points we know that the maximum must occur at one or other of the endpoints. Noting that $m(1) = m(144) = 145,$ we have established a range for $m$ of $24$ to $145,$ a range which includes $145 - 4 + 1 = \boxed{122}$ integer values.

Alternately, we could have used the A.M.-G.M. inequality to show that

$m(y) = \dfrac{144}{y} + y \ge 2\sqrt{\dfrac{144}{y}*y} = 24,$

which is achieved when $y = 144^{\cos^{2}(x)} = 12 \Longrightarrow x = \dfrac{\pi}{4} + \dfrac{n\pi}{2}.$

We could then make the qualitative observation that $m(y)$ increases as (positive) $y$ goes both to the left and to the right of this minimum, so any maximum must occur at the endpoints.

Brian has written a delightfully clear solution, as usual. For the sake of variety, let me submit my own solution, based on concavity.

Let $a=\sin^2(x)$ and $b=\cos^2(x)$ , with $a+b=1$ . Since the graph of $f(t)=144^t$ is concave up, we have the following inequalities for $m=f(a)+f(b)$ : $2f\left(\frac{a+b}{2}\right)\leq{f(a)+f(b)}\leq{f(0)+f(a+b)}$ or $24\leq{m}\leq{145}$ . Since $m$ is a continuous function of $x$ , all the intermediate integers are attained as well, so that the overall number is $145-24+1=\boxed{122}$