Final answer:
The expression (-tan²x)/(sec²x-1) simplifies to -1 by using trigonometric identities, specifically the Pythagorean identity tan²x + 1 = sec²x.
Step-by-step explanation:
To simplify the expression (-tan²x)/(sec²x-1), we need to use trigonometric identities. Recall that tan²x = sin²x/cos²x and sec²x = 1/cos²x. We also know the Pythagorean identity tan²x + 1 = sec²x, which implies tan²x = sec²x - 1. By substituting these identities into the original expression, we obtain the following:
- tan²x = sin²x/cos²x
- sec²x - 1 = tan²x
- Our expression simplifies to (-tan²x)/tan²x
After substituting, we are simply left with -1, as the tan²x in the numerator and denominator cancel each other out.
Therefore, the simplified form of the given expression (-tan²x)/(sec²x-1) is -1.