$b (c c o s A - a c o s C) = c^{2} - a^{2}$

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Solution

$b (c c o s A - a c o s C) = c^{2} - a^{2} R H S = c^{2} - a^{2} = k^{2} s i n^{2} C - k^{2} s i n^{2} A = k^{2} (s i n^{2} C - s i n^{2} A) = k^{2} s i n (C + A) . s i n (C - A) = k^{2} s i n (π - B) . s i n (C - A) = k^{2} s i n B . s i n (C - A) = k s i n B k s i n (C - A) = b k s i n (C - A) = b k (s i n C . c o s A - s i n A . c o s C) = b (k s i n C . c o s A - k s i n A . c o s C) = b (c c o s A - a c o s C) = L H S$

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b (c cos A−a cos C)=c2−a2

b (c cos A−a cos C)=c2−a2RHS=c2−a2=k2 sin2 C−k2 sin2 A=k2 (sin2 C−sin2 A)=k2 sin (C+A). sin(C−A)=k2 sin (π−B). sin(C−A)=k2 sin B . sin(C−A)=k sin B k sin(C−A)=bk sin (C−A)=bk (sin C.cos A−sinA . cosC)=b (k sin C.cos A−k sinA. cosC)=b (c cos A−a cos C)=LHS

$b (c c o s A - a c o s C) = c^{2} - a^{2}$