Let $A$ be a normal linear operator on a complex finite dimensional unitary space which satisfies the equation $ \tan A + \cot A = 2I$

Question

Let $A$ be a normal linear operator on a complex finite dimensional unitary space which satisfies the equation: $$ \tan A + \cot A = 2I$$ Prove that $A$ has to be hermitian. I tried by looking at the eigenvalues of the operator $ f(A) $ where $f$ is the function $ f(\lambda)=\tan \lambda+ \cot\lambda $ and using that the eigenvalues of a hermitian operator have to be real. The only eigenvalue of $f(A)$ is then $2$? And then the eigenvalues of $A$ would be $ \pi /4 + k*\pi$?