Final answer:
The given trigonometric equation secx tanx(1 - sin²x) = sinx simplifies to sinx by using the Pythagorean identity and properties of secx and tanx. The equation is therefore an identity that always holds true where defined, making option c) the correct answer.
Step-by-step explanation:
When we analyze the trigonometric equation secx tanx(1 - sin²x) = sinx, we need to determine if this is an identity and why. First, recall that secx is the reciprocal of cosx, so secx = 1/cosx, and tanx is sinx/cosx. Multiplying these together gives us sinx, which is the numerator of tanx.
We know that (1 - sin²x) is equivalent to cos²x according to the Pythagorean identity. Thus, when we multiply secx * tanx by (1 - sin²x), it simplifies to sinx:
- secx tanx (1 - sin²x) = (1/cosx) * (sinx/cosx) * cos²x = sinx.
Therefore, the given equation simplifies to a known identity, which is true for all values of x where secx and tanx are defined. Hence, option c) It simplifies to a known identity is the correct answer.