In any triangle, if \frac {\cos A+2\cos C}{\cos A+2\cos B}=\frac

luminarc24lry

luminarc24lry

Answered question

2022-04-24

In any triangle, if cosA+2cosCcosA+2cosB=sinBsinC , prove that the triangle is either isosceles or right angled.
My Attempt:
Given:
cosA+2cosCcosA+2cosB=sinBsinC
b2+c2a22bc+2a2+b2c22abb2+c2a22bc+2a2+c2b22ac=bc
On simplification,
ab2+ac2a3+2a2c+2b2c2c3ab2+ac2a3+2a2b+2bc22b3=bc

Answer & Explanation

Aliana Sexton

Aliana Sexton

Beginner2022-04-25Added 20 answers

Alternate approach:
Cross multiply
cosAsinC+sin2C=sinBcosA+sin2B
cosA(sinCsinB)(sin2Bsin2C)=0
cosA2sinCB2cosB+C22sin(BC)cos(B+C)=0
cosA2sinCB2cosB+C2+2sin(BC)cosA=0
2cosA[sinCB2cosB+C2+2sinCB2cosBC2]=0
2cosAsinBC2[cosB+C22cosBC2]=0
IF: cosB+C22cosBC2=0tanB2tanC2=13

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