f sides of a tringle a,b,c are in AP then prove that :

COSA.COTA/2, COSB.COTB/2, COSC.COTC/2 ARE IN AP.

37 minutes ago

Answers : (1)

 

cot A/2= (cos A/2)/(sin A/2) = 2 cos^2A/2/sin A = (cos A+1)/ Sin A

Therfore,

Visulaizing 1st term = cos^2 A/sin A + cos A/sin A = (1-sin ^2A)/sin A+ cos A/sin A

= (1+ cos A)/sin A - sin A

so ,as sin A sin B and sin C are in AP we need to show that

(1+cos A)/Sin A, (1+ cos B)/sin B, (1+ cos C)/sin C are in AP

1+ cos A = 1+ (b^2+c^2-a^2)/2bc

= (2bc+b^2+c^2-a^2)/2bc

so (1+cos A)/sin A = (2bc +( b^2+c^2-a^2)/2abc k

as sin A/ a= sin B/b = sin C/c = k

(1+Cos C)/sin C = (2ab+(b^+a^2-c^2)/2abck

so, (1+cos A)/Sin A, (1+ cos B)/sin B, (1+ cos C)/sin C are in AP

Now,as sin A, sin B sin C are in AP and also (1+cos A)/Sin A, (1+ cos B)/sin B, (1+ cos C)/sin C are in AP, so cosA.cotA/2,cosB.cotB/2, cosC.cotC/2 are in AP (Hence proved).