S : Area of AB^Δ C in AB^Δ C : (a^( 2) /(4S)) =^? (1/2) (cot(B)+cot(C)) (a^( 2) /(4S)) = (( a^( 2) )/(4 ((1/2) bc sin(A))))=((4R^2 sin^( 2) (A))/(8R^( 2) sin(B)sin(C)sin(A))) = ((sin (A ))/(2sin(B)sin(C))) =^(A+B+C=π) ((sin(B)cos(C)+cosBsin(C))/(2sin(B)sin(C))) = (1/2) (cot(B)+cot(C)) ■


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Question Number 201134 by mnjuly1970 last updated on 30/Nov/23

$S : A r e a o f A \overset{Δ}{B C}$ $i n A \overset{Δ}{B C} : \frac{a^{2}}{4 S} \overset{?}{=} \frac{1}{2} (c o t (B) + c o t (C))$ $\frac{a^{2}}{4 S} = \frac{a^{2}}{4 (\frac{1}{2} b c s i n (A))} = \frac{4 R^{2} {s i n}^{2} (A)}{8 R^{2} s i n (B) s i n (C) s i n (A)}$ $= \frac{s i n (A)}{2 s i n (B) s i n (C)}$ $\overset{A + B + C = π}{=} \frac{s i n (B) c o s (C) + c o s B s i n (C)}{2 s i n (B) s i n (C)}$ $= \frac{1}{2} (c o t (B) + c o t (C)) ◼$
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