Final answer:
The expression tan²(x) - sec²(x) simplifies to -1 after substituting the trigonometric identities for tan(x) and sec(x) in terms of sine and cosine and using the identity sin²(x) + cos²(x) = 1.
Step-by-step explanation:
To simplify the trigonometric expression tan²(x) - sec²(x) and write it in terms of sine and cosine, we can use trigonometric identities. Recall that tan(x) = sin(x)/cos(x) and sec(x) = 1/cos(x), hence tan²(x) = sin²(x)/cos²(x) and sec²(x) = 1/cos²(x). substituting these into the expression, we get:
sin²(x)/cos²(x) - 1/cos²(x) = (sin²(x) - 1) / cos²(x).
Using another identity, sin²(x) + cos²(x) = 1, we can replace sin²(x) with 1 - cos²(x). Substituting this into our expression gives us:
(1 - cos²(x) - 1) / cos²(x) = -cos²(x) / cos²(x) = -1,
which is the simplified form of the original trigonometric expression in terms of sine and cosine.