Can anyone prove this please?

#{(b^2-c^2)//a^2} xx sin 2A + {(c^2-a^2)//b^2}xx sin 2B + {(a^2-b^2)//c^2} xx sin 2C# =0

3 Answers
Feb 20, 2018

Use the sine law for triangles and some simple trigonometric identities.

Explanation:

From the sine law of triangles

#a/{sin A}= b/{sin B} = c/{sin C}#

we can easily see that

#{b^2 -c^2}/a^2 = {sin^2B-sin^2C}/sin^2A= {(sin B-sinC)(sin B+ sin C)}/{sin^2A} = {2 sin ({B-C}/2)cos({B+C}/2)times 2 sin({B+C}/2)cos({B-C}/2)}/sin^2A = {sin(B-C)sin(B+C)}/sin^2A = {sin(B-C)sin(pi-A)}/sin^2A = sin(B-C)/sinA#

So that

# {b^2 -c^2}/a^2 times sin2A = 2cosAsin(B-C) = 2 cosAsinBcosC-2cosAcosBsinC#

The other two terms can be obtained from this one by simply cyclically permuting #A#, #B# and #C#. Adding the three terms leads to the proof trivially .

Feb 20, 2018

Please see below.

Explanation:

The first term of #LHS=(b^2-c^2)/a^2*sin2A#

#=(4R^2[sin^2A-sin^2B])/(4R^2*sin^2A)*sin2A#

#=(sin(B+C)sin(B-C))/sin^2A*sin2A#

#=(sinAsin(B-C))/(sinA*sinA)*2sinA*cosA#

#=2cosAsin(B-C)#

#=sin(A+B-C)-sin(A-B+C)#

#=sin(pi-2C)-sin(pi-2B)=sin2C-sin2B#

Similarly The second term#=sin2A-sin2B# and

The third term#=sin2B-sin2A#

Whole #LHS=sin2C-sin2B+sin2A-sin2C+sin2B-sin2C=0#

Note that #sin^2A-sin^2B=sin(A+B)*sin(A-B)#

Feb 20, 2018

Kindly refer to the Explanation.

Explanation:

Prerequisites : In the usual Notation for #DeltaABC,#

Sine-Rule : #a/sinA=2R, or, sinA=a/(2R)#.

Cosine-Rule : #cosA=(b^2+c^2-a^2)/(2bc)#.

We have, #(b^2-c^2)/a^2*sin2A=(b^2-c^2)/a^2*(2sinAcosA)#,

#=(b^2-c^2)/a^2*{2*a/(2R)*(b^2+c^2-a^2)/(2bc)}#,

#={(b^2-c^2)(b^2+c^2-a^2)}/(Rabc)#,

#={(b^2-c^2)(b^2+c^2)-a^2(b^2-c^2)}/(Rabc)#,

#rArr(b^2-c^2)/a^2*sin2A={(b^4-c^4)-a^2(b^2-c^2)}/(Rabc)#.

Obtaining similar expressions for the remaining terms of the left

member and adding them, the result follows.