Question

Verify the identity. csc(x) cos²(x) sin(x) = csc(x)

A) cos(x)
B) sin(x)
C) cot(x)
D) sec(x)

Answer 1

The identity csc(x) cos²(x) sin(x) = csc(x) cannot be verified because simplifying the left side leads to cos²(x), not csc(x). The provided identity is incorrect, as there is an extra term cos(x) on the left which does not match any given option.

Step-by-step explanation:

To verify the identity csc(x) cos²(x) sin(x) = csc(x), we need to simplify the left side of the equation. The cosecant function is defined as csc(x) = 1/sin(x). When we have cos²(x), it refers to (cos(x))², and we can use the trigonometric identity sin²(x) + cos²(x) = 1 to relate sine and cosine.

Let's simplify the left side of the identity:

1. Replace csc(x) with its definition, which gives us 1/sin(x) × cos²(x) × sin(x).
2. The sin(x) in the numerator and the denominator cancel each other out, leaving us with cos²(x).
3. Recognizing that cos²(x) can be rewritten as (1 - sin²(x)) from the Pythagorean identity.
4. This simplification leads us to 1 × (1 - sin²(x)) = 1 - sin²(x), which simplifies to cos²(x) again based on the Pythagorean identity.

So, this verifies that the left side equals csc(x), but with an extra term cos(x) that does not appear on the right side. Thus, the identity provided is incorrect; there is no match from the given multiple-choice options that satisfies the identity as stated. The correct identity without the cos(x) term should be simply csc(x) sin(x) = 1.