Can we subtract a trigonometric term from a polynomial? Can we find the root of a function like $f(x) = x^2-\cos(x)$ using accurate algebra or do we need to resort to numerical methods approximations?
thanks.

The answer to your posed problem cannot be expressed in general in terms of elementary formulas in closed form. So in practice, numerical root finding is the only way to go. Luckily, for your question finding a root approximately is not that hard using Newton's method.
