To find csc(x) given the equation sin(x) * cot(x) = cos(x) * √3, we can start rewriting the equation using trigonometric identities.
First, recall the identity for cot(x): cot(x) = cos(x) / sin(x). Thus, we can substitute it into our equation:
sin(x) * (cos(x) / sin(x)) = cos(x) * √3
This simplifies to:
cos(x) = cos(x) * √3
Assuming cos(x) ≠ 0, we can divide both sides by cos(x):
1 = √3
However, this is not true, indicating that either cos(x) = 0 or there is no solution under these conditions. If we set cos(x) = 0, then we are looking for the values of x where cosine equals zero, specifically where:
x = π/2 + nπ, where n is any integer.
Now, given that csc(x) is the reciprocal of sin(x), we can find values for sin(x) at these points:
- For x = π/2: sin(π/2) = 1, so csc(π/2) = 1.
- For x = 3π/2: sin(3π/2) = -1, so csc(3π/2) = -1.
Thus, csc(x) can take on values of 1 and -1 at these angles.
To summarize:
- If cos(x) = 0, then csc(x) = 1 or csc(x) = -1.
- Given the initial equation does not lead to valid cosine results, the only valid solutions come from cos(x) = 0.
Therefore, we conclude that the possible values for csc(x) are 1 and -1.