We can look that up; one place is https://mathworld.wolfram.com/Multiple-AngleFormulas.html .  {For brevity let S=sin(x)}  sin(5x) = 5S-20S^3+16S^5.

Now onto the actual problem: evaluate this at x=18 degrees.

Then 1 = 5S-20S^3+16S^5.  

Then subtract 1 from each side and factor: (S-1)*(4S^2+2S-1)^2 = 0.  

We know sin(18) = S != 1, so S must be one of the roots of 4S^2+2S-1 = 0.

The roots are S=(sqrt(5)-1)/4 and S=(-sqrt(5)-1)/4.  The second one is negative so it is rejected since we know that sin(18) is between 0 and 1.  Therefore sin(18) = (sqrt(5)-1)/4.

Now we can do what K Sengupta did and just directly evaluate the expression S^3+S^2 at (sqrt(5)-1)/4 to get 1/8.