Final answer:
The expression simplifies to -csc^2(x) using trigonometric identities and properties of even and odd functions. Comparing this result with the given equation, the correct answer is d) csc(x).
Step-by-step explanation:
The student is asking to simplify an expression involving trigonometric functions and to verify whether it equates to a given trigonometric identity. The expression given is (csc(-x) / sec(-x)) * (cos(-x) / sin(-x)) = -2 * cot(x), and we need to find which option (a, b, c or d) it simplifies to.
To start, we will use the fundamental trigonometric identities. These identities are:
- csc(x) = 1 / sin(x)
- sec(x) = 1 / cos(x)
- cos(-x) = cos(x) (cosine is even)
- sin(-x) = -sin(x) (sine is odd)
- cot(x) = cos(x) / sin(x)
Using these identities, we can rewrite the original expression:
((1/sin(-x)) / (1/cos(-x))) * (cos(-x) / sin(-x))
Now, simplifying and using the properties of the trigonometric functions being even or odd, we have:
(cos(x) / -sin(x)) * (cos(x) / sin(x)) = cos^2(x) / (-sin^2(x)) = -cot^2(x)
Now, by comparing this result with the original expression on the right side -2 * cot(x), we can see that it doesn't match. However, we have not used the identity cot^2(x) + 1 = csc^2(x). With this identity, we can write:
-cot^2(x) = -1 + 1 - cot^2(x) = -(csc^2(x) - 1)
This means our expression simplifies to -csc^2(x), so the correct answer is option d) csc(x).