In a test a student simplifies ${\tan }^{2}A-{\sec }^{2}A$ and writes 1.
What is the correct value?

The identity is ${\sec }^{2}A-{\tan }^{2}A=1$. Reversing the order of subtraction reverses the sign, so ${\tan }^{2}A-{\sec }^{2}A=-1$. This is a pure sign slip, not a conceptual error. Check at $A=45$ deg: ${\tan }^{2}45^{\circ }-{\sec }^{2}45^{\circ }=1-2=-1$.