Is it possible (not numerical) to find that? $ x $ such as:

$$

tan (x) + cos (x) = 1/2

$$

?

All my attempts are finished in a 4-degree polynomial. Example: Calling c = cos (x):

$$

frac { sqrt {1-c ^ 2}} {c} + c = frac {1} {2}

$$

$$

sqrt {1-c ^ 2} + c ^ 2 = frac {1} {2} c

$$

$$

1-c ^ 2 = c ^ 2 ( frac {1} {2} -c) ^ 2 = c ^ 2 ( frac {1} {4} -c + c ^ 2)

$$

$$

c ^ 4-c ^ 3 + frac {5} {4} c ^ 2-1 = 0

$$