Let's find the implicit derivative:
[tex]\begin{gathered} \frac{d}{dx}\tan ^2y=\frac{d}{dx}(x) \\ \sec ^2y\frac{dy}{dx}=1 \\ \frac{dy}{dx}=\frac{1}{\sec^2y} \\ \frac{dy}{dx}=\cos ^2y \end{gathered}[/tex]
Therefore:
[tex]\frac{dy}{dx}=\cos ^2y[/tex]
Since the derivative of y is equal to the cosine and the cosine is a continous function for all the real numbers then the derivative of y is defined for all values of y.
