can anyone help me explain this

[calcnoob]

ok i have this problem where i have to find the x so that i get the maximum viewing angle

and i found an equation to solve it which is arctan(10/x)-arctan(3/x)=beta and then i derive the equation and set it to zero to find the maximum and it helped on my exam but i want to know why im guessing it has something to
do with symmetric triangles

can someone help me explain why this is the equation

 

[HallsofIvy]

Let \(\displaystyle \phi\) be the angle just below \(\displaystyle \theta\). Then \(\displaystyle tan(\phi)= \frac{3}{x}\) and \(\displaystyle tan(\theta+ \phi)= \frac{13}{x}\). Now use the trig identity \(\displaystyle tan(\theta+ \phi)= \frac{tan(\theta)+ tan(\phi)}{1+ tan(\theta)tan(\phi)}\).

If you don't recognise that identity you can derive it from \(\displaystyle sin(x+y)= sin(x)cos(y)+ cos(x)sin(y)\), \(\displaystyle cos(x+y)= cos(x)cos(y)- sin(x)sin(y)\) and, of course, \(\displaystyle tan(x)= \frac{sin(x)}{cos(x)}\).