Final answer:
Option D. The equation y = sin(x) / (1 + cos(x)) simplifies to (1 - cos(x)) / sin(x), which further simplifies to csc(x) - cot(x). None of the options given match this expression exactly, but the closest single function listed is csc(x).
Step-by-step explanation:
To solve the equation y = sin(x) / (1 + cos(x)), we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to rewrite sin(x) in terms of cos(x). By subtracting cos^2(x) from both sides, we get sin^2(x) = 1 - cos^2(x). Therefore, sin(x) can be expressed as √(1 - cos^2(x)). When we substitute this back into the original equation, we see that it does not match with the options given (a) cot(x), (b) tan(x), (c) sec(x), or (d) csc(x). However, we can simplify the original expression by employing a trigonometric identity.
Using the trigonometric identity tan(x) = sin(x)/cos(x), we can multiply the numerator and the denominator of the original equation by 1 - cos(x). This process yields:
- y = sin(x) / (1 + cos(x))
- y = sin(x)(1 - cos(x)) / (1 - cos^2(x))
- y = sin(x)(1 - cos(x)) / sin^2(x) (Since 1 - cos^2(x) = sin^2(x))
- y = (1 - cos(x)) / sin(x)
- y = 1/sin(x) - cos(x)/sin(x)
- y = csc(x) - cot(x)
Therefore, none of the given options exactly match the simplified form of the equation. However, if we consider the terms separately, csc(x) is part of the solution, though it is not the entire solution. The closest single trigonometric function in the options to the simplified form is Option d: y = csc(x), but it should be noted that this is only part of the simplified expression, and the full expression also includes - cot(x).