How do I find the fundamental period of the given signal?
If your top equation is really 
$$
x(t) = 2\cos\left(\frac 45 \pi t\right)\sin^2\left(\frac{16}{3} t\right)\tag{1}
$$
You gonna have a hard time getting the fundamental period/frequency as the there isn't an exact integer relating the two periods/frequencies. You can make approximations of the $\pi$ multiplier/divisor but the errors accumulates and this doesn't cut it. However if your equation is:
$$
x(t) = 2\cos\left(\frac 45 \pi t\right)\sin^2\left(\frac{16}{3} \color{magenta}\pi t\right)\tag{2}
$$
You could then easily extract the fundamental frequency $f_0$, and simply get the inverse $\displaystyle\frac{1}{f_0}$  to get the fundamental period $T_0$. For a signal $x(t)$

$$
x(t) = \sum_{k=1}^N a_k\cos\left(2\pi f_k t+ \phi_k\right)
$$

The fundamental frequency is the greatest common divisor (GCD) of all the frequencies; or 
$$\mathrm{GCD}\left\{f_k\right\}\quad\big\vert \quad k = 1, \ldots, N$$

With non-integer values of $f_k$, you get the frequencies to the same common denominator or their least common multiple (LCM) and $f_0$ is the GCD of the numerators over this LCM, their common denominator. In your case, you have the form: 
$$
x(t) = \prod_{k=1}^N a_k\cos\left(2\pi f_k t+ \phi_k\right)
$$
But the same principle applies. So, looking at the second-last equation you have two distinct frequencies: $\displaystyle f_1 = \frac 25$ and $\displaystyle f_2 =\frac{16}{3}$. 

So in the end, you get $\displaystyle f_0 = \frac{2}{15}\implies T_0 = \frac 1f_0 = 7.5$.

Below is a plot illustrating this with the four waveforms below:

\begin{cases}
\color{cyan}{y_1} &amp;= 2\cos\left(\frac 45 \pi t\right)\\
\color{green}{y_2} &amp;= \sin^2\left(\frac{16}{3}\pi t\right)\\
\color{blue}{y_3} &amp;= 2\cos\left(\frac 45\pi t\right)\cdot \sin^2\left(\frac{16}{3}\pi t\right)\\
\color{red}{y_4} &amp;= 2\cos\left(2\pi\frac{2}{15}t\right)\\
\end{cases}


You see that the fundamental period of your signal $y_3$ ($\color{blue}{\rm blue}$) is at exactly $7.5 \ \rm seconds$ in the plot, matching exactly that of a pure sinusoidal wave $y_4$ ($\color{red}{\rm red}$) at $\displaystyle f_0 = \frac{2}{15}$.