prove using sine rule - 1) sin ( B-C)/2 = b-c/a cos ( A/2) 2)1+cos(A-B) cos C / 1+cos(A-C) cos - Maths - Trigonometric Functions

Question

prove using sine rule - 1) sin ( B-C)/2 = b-c/a cos ( A/2)
2)1+cos(A-B) cos C / 1+cos(A-C) cos B = a2+ b2/ a2- c2
3) (b-c)cot (A/2) + (c-a)cot (B/2) + (a-b)cot (C/2) = 0
4) a cos(B-C/2) = (b+c) sin(A/2)
5) c/(a-b) = tan(A/2) + tan(B/2) / tan(A/2) - tan(B/2)
6) a(cos C - cos B) = 2(b-c) cos^2 (A/2)

Santosh Kumar · Accepted Answer

Due to paucity of time it would not be possible for us to provide the answers to all your queries. We are providing solutions to some your good queries. Try solving rest of the questions yourself and if you face any difficulty then do get back to us.

(1) Let $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k$

Then, a = k sin A, b = k sin B, c = k sin C

$R.H.S = (\frac{b - c}{a}) \cos \frac{A}{2} = (\frac{k \sin B - k \sin C}{k \sin A}) \cos \frac{A}{2} = (\frac{\sin B - \sin C}{\sin A}) \cos \frac{A}{2} = \frac{2 \sin (\frac{B - C}{2}) \cdot \cos (\frac{B + C}{2})}{2 \sin \frac{A}{2} \cdot \cos \frac{A}{2}} \cdot \cos \frac{A}{2} = \frac{\sin (\frac{B - C}{2}) \cdot \cos (\frac{π - A}{2})}{\sin \frac{A}{2}} = \frac{\sin (\frac{B - C}{2}) \cdot \cos (\frac{π}{2} - \frac{A}{2})}{\sin \frac{A}{2}} = \frac{\sin (\frac{B - C}{2}) . \cdot \sin \frac{A}{2}}{\sin \frac{A}{2}} = \sin (\frac{B - C}{2}) = L . H . S$

(4) $R.H.S = (b + c), \sin \frac{A}{2}$

$= (k \sin B + k \sin C) \cdot \sin \frac{A}{2} = k (\sin B + \sin C) \cdot \sin \frac{A}{2} = k \times 2 \sin (\frac{B + C}{2}) \cdot \cos (\frac{B - C}{2}) \cdot \sin \frac{A}{2} = k (2 \sin \frac{A}{2} \cdot \cos \frac{A}{2}) \cdot \cos (\frac{B - C}{2}) = k \sin A \cdot \cos (\frac{B - C}{2}) = a \cos (\frac{B - C}{2}) = L.H.S$

https://www.meritnation.com/ask-answer/question/prove-using-sine-rule-a-cos-c-cos-b-2-b-c/trigonometric-functions/2522048

https://www.meritnation.com/ask-answer/question/prove-using-sine-rule-1-cos-a-b-cos-c-1-cos-a-c/trigonometric-functions/2522056

prove using sine rule - 1) sin ( B-C)/2 = b-c/a cos ( A/2) 2)1+cos(A-B) cos C / 1+cos(A-C) cos B = a2+ b2/ a2- c2 3) (b-c)cot (A/2) + (c-a)cot (B/2) + (a-b)cot (C/2) = 0 4) a cos(B-C/2) = (b+c) sin(A/2) 5) c/(a-b) = tan(A/2) + tan(B/2) / tan(A/2) - tan(B/2) 6) a(cos C - cos B) = 2(b-c) cos^2 (A/2)