Question

a d c b bret kamala sin(c) = c/a sin(c) = a/a sin(a) = a/a sin(a) = a/c a = c/sin(a) a sin(c) = a a = c/sin(c) c sin(a) = a c/sin(a) = c/sin(c) a sin(c) = c sin(a) sin(c)/c = sin(a)/a sin(c)/c = sin(a)/a the proof was correctly completed by both bret and kamala the next step in the proof is to write an expression for the cosine of a and write an expression for the cosine of b the sine of b the length a

Explanation:

Step1: Analyze the sine - rule proof

Both Bret and Kamala start with the basic sine - rule relationships $\sin(C)=\frac{c}{A}$ and $\sin(A)=\frac{a}{A}$ (assuming $A$ is the hypotenuse of a right - angled triangle related to the angles $A$ and $C$ in a larger geometric context). They then manipulate these equations to get to the sine - rule form $\frac{\sin(A)}{a}=\frac{\sin(C)}{c}$.

Step2: Determine the next step

In a triangle, after establishing the sine - rule, a common next step in a more comprehensive proof involving angles and sides is to consider the cosine of an angle. The cosine of an angle can be related to the sides of a triangle using the cosine - rule, and it helps in further analyzing the properties of the triangle.

Answer:

The proof was correctly completed by: Both Bret and Kamala
The next step in the proof is to: write an expression for the cosine of A