Give the reason that justifies each step in the following pr

Give the reason that justifies each step in the following proof of the Law of Cosines. bd = (a + c)e bd = (a + c)(a - c) b(2a cos C - b) = (a + c)(a - c) 2ab cos C - b^2 = a^2 -c^2 c^2 = a^2 + b^2 - 2ab cos C

Solution

bd=(a+c)e ( by chord intersection theorem)

and from diagram, e=a-c

Therefore bd=(a+c)(a-c)

And in right triangle, cos theta= adjacent/hypotenuse

here adjacent= b+d

and hypotenuse= a+a=2a

Therefore cos C= b+d/2a

b+d=2acos C

d=2a cos C - b

Therefore

b(2acosC - b)= a^2 - c^2

distributing b

2abcos C - b^2=a^2 - c^2

c^2= a^2 + b^2 - 2abcos C