CBSE Class 11-science Answered

Let cos A + cos  B + cos C = a

And sin A + sin B + sin C = b

To prove a =b = 0

We can prov that a² + b² = 0.

Now (cos A + cos B + cos C)²

= cos² A + cos² B + cos² C + 2 (cos A cos B + cos B cos C + cos C cos A)      …(i)

and (sin A + sin B + sin C)² 

= sin² A + sin ² B  + sin² C + 2 (sin A sin B + sin B sin C + sin C sin A).

Given cos (B-C) + cos (A-B) + cos (C-A) = -3/2                                         …(ii)

i.e., cos B cos C + sin B sin C + cos A cos B + sin A sin B + cos C cos A

+ sin C sin A = -3/2                                                                               …(iii)

Adding (i) and (ii)

(cos² A + sin² A) + (cos² B + sin² B) + (sin² C + cos² C) +  

2(cos B cos C + sin B sin C + cos A cos B + sin A sin B + cos C cos A + sin C sin A)

3 + 2 x -3/2 = o      (using (iii))

So a² + b² = 0

Þ a = 0 and b = 0.

i.e., cos A + cos B + cos C = sin A + sin B + sin C = 0.

Regards

Team Topperlearning