Trigonometric Identities · Superchallenge problem

For every angle $x$ at which all four terms are defined, the expression $\dfrac{1}{1 + \sin^2 x} + \dfrac{1}{1 + \cos^2 x} + \dfrac{1}{1 + \sec^2 x} + \dfrac{1}{1 + \csc^2 x}$ has the same value. Find it.