Final answer:
To show that y = sec^2x and y = tan^2x have the same derivative, differentiate both expressions with respect to x. The derivatives of both expressions are equal to 2sec^2x * tanx or 2tanx * sec^2x. The expression y = sec^2x − tan^2x simplifies to y = sec^2x.
Step-by-step explanation:
To show that y = sec^2x and y = tan^2x have the same derivative, we need to differentiate both expressions with respect to x. First, let's find the derivative of y = sec^2x:
y' = d/dx(sec^2x) = 2sec^2x * tanx
Next, let's find the derivative of y = tan^2x:
y' = d/dx(tan^2x) = 2tanx * sec^2x
As we can see, the derivatives of both expressions are equal to 2sec^2x * tanx or 2tanx * sec^2x.
Now, let's consider the expression y = sec^2x − tan^2x. We can simplify this expression by applying the identity 1 − tan^2x = sec^2x:
y = sec^2x − tan^2x = 1 − tan^2x = sec^2x
Therefore, y = sec^2x − tan^2x simplifies to y = sec^2x. In other words, they are the same function.