Using cosine rule prove the sine rule

Sure, here is the step by step proof of the sine rule using the cosine rule:

Step 1: Consider a triangle ABC with sides a, b, c and angles A, B, C opposite to them respectively.

Step 2: According to the cosine rule, we have:

c² = a² + b² - 2ab cos(C)

Step 3: Rearrange the equation to isolate cos(C):

cos(C) = (a² + b² - c²) / 2ab

Step 4: We know that cos(C) = sqrt(1 - sin²(C)). Substitute this into the equation:

sqrt(1 - sin²(C)) = (a² + b² - c²) / 2ab

Step 5: Square both sides to get rid of the square root:

1 - sin²(C) = (a² + b² - c²)² / 4a²b²

Step 6: Rearrange the equation to isolate sin²(C):

sin²(C) = 1 - (a² + b² - c²)² / 4a²b²

Step 7: We know that sin²(C) = (sin(C))². Substitute this into the equation:

(sin(C))² = 1 - (a² + b² - c²)² / 4a²b²

Step 8: Take the square root of both sides:

sin(C) = sqrt[1 - (a² + b² - c²)² / 4a²b²]

Step 9: We know that sin(C) = c / 2R, where R is the circumradius of the triangle. Substitute this into the equation:

c / 2R = sqrt[1 - (a² + b² - c²)² / 4a²b²]

Step 10: Rearrange the equation to isolate c:

c = 2R sqrt[1 - (a² + b² - c²)² / 4a²b²]

Step 11: This is the sine rule, which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.