Prove that A 2 - B 2 - C 2 A - B - C - a-b-c 2 a 2 - b 2 - c 2

A2+ B2 + C2 A+ B+ C= √(8(8 a)(8 b)(8 c)) + √(8(8 b)(8 a)(8 c)) + √(8(8 c)(8 a)(8 b))[ b^2+c^2 a^22bc2 bc ] +[ a^2+c^2 b^22ac2ac] + [ [ a^2+ b^2 c^22ab2 ...

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Standard XII

Mathematics

Sin(A+B)Sin(A-B)

Prove that A...

Question

Prove that
$\frac{cot \frac{A}{2} + cot \frac{B}{2} + cot \frac{C}{2}}{cot A + cot B + cot C} = \frac{{(a + b + c)}^{2}}{a^{2} + b^{2} + c^{2}}$

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Solution

$\frac{cot \frac{A}{2} + cot \frac{B}{2} + cot \frac{C}{2}}{cot A + cot B + cot C} = \frac{\sqrt{\frac{8 (8 - a)}{(8 - b) (8 - c)}} + \sqrt{\frac{8 (8 - b)}{(8 - a) (8 - c)}} + \sqrt{\frac{8 (8 - c)}{(8 - a) (8 - b)}}}{⎡ ⎢ ⎢ ⎢ ⎣ \frac{\frac{b^{2} + c^{2} - a^{2}}{2 b c}}{\frac{2 △}{b c}} ⎤ ⎥ ⎥ ⎥ ⎦ + ⎡ ⎢ ⎢ ⎢ ⎣ \frac{\frac{a^{2} + c^{2} - b^{2}}{2 a c}}{\frac{2 △}{a c}} ⎤ ⎥ ⎥ ⎥ ⎦ + ⎡ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎣ \frac{\frac{a^{2} + b^{2} - c^{2}}{2 a b}}{\frac{2 △}{a b}} ⎤ ⎥ ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎥ ⎦}$
using $cot \frac{A}{2}, cos A, sin A$ expansion
$= \frac{\frac{8 (8 - a) + 8 (8 - b) + 8 (8 - c)}{△}}{[\frac{b^{2} + c^{2} - a^{2}}{4 △}] + [\frac{a^{2} + c^{2} - b^{2}}{4 △}] + [\frac{a^{2} + b^{2} - c^{2}}{4 △}]}$
$= \frac{48^{2}}{(a^{2} + b^{2} + c^{2})} = \frac{(a + b + c)^{2}}{(a^{2} + b^{2} + c^{2})}$

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Prove thatcotA2+cotB2+cotC2cotA+cotB+cotC=(a+b+c)2a2+b2+c2

Prove that
$\frac{cot \frac{A}{2} + cot \frac{B}{2} + cot \frac{C}{2}}{cot A + cot B + cot C} = \frac{{(a + b + c)}^{2}}{a^{2} + b^{2} + c^{2}}$