cotxsinx=cosx
cotx=cosx/sinx
Therefore, (cosx/sinx)sinx=cosx
cosx=cosx
To solve the equation cot(x)sin(x) = cos(x), we can use trigonometric identities.
1. Rewrite cot(x) as 1/tan(x). The equation becomes (1/tan(x))sin(x) = cos(x).
2. Since sin(x)/cos(x) = tan(x), we can substitute tan(x) for sin(x)/cos(x) in the equation. This gives us (1/tan(x))(sin(x)/cos(x)) = cos(x).
3. Multiply both sides of the equation by cos(x) to eliminate the denominators. The equation becomes sin(x) = cos^2(x).
4. Use the identity sin^2(x) + cos^2(x) = 1. Rewrite the equation as 1 - cos^2(x) = cos^2(x).
5. Simplify the equation by combining like terms. We have 1 = 2cos^2(x).
6. Divide both sides of the equation by 2. The equation becomes 1/2 = cos^2(x).
7. Find the square root of both sides of the equation. We have ±√(1/2) = cos(x).
Therefore, the solutions for x are x = π/4 ± (2nπ), where n is an integer.
To summarize, the solutions for cot(x)sin(x) = cos(x) are x = π/4 ± (2nπ), where n is an integer.