Question

A+B+C=180 and cosA=cosB.cosC. prove by tanB. tanC=2

Answer 1

a=180-(b+c) so 
cosa = - cos(b+c) = - (cosb cosc - sinb sinc) = - cosb cosc + sinb sinc 
But cosa =cosb cos c so 
- cosb cosc + sinb sinc = cosb cos c and therefore 
sinb sinc = 2 cosb cosc 
Dividing through by cosb cosc we get 
tanb tanc = 2 q.e.d

Answer 2

a=180-(b+c)   So,

cosA = - cos(B+C) = - (cosB cosC - sinB sinC) = - cosB cosC + sinB sinC

But cosA =cosB cosC so

- cosB cosC + sinB sinC = cosB cosC and therefore

sinB sinC = 2 cosB cosC

Dividing through by cosB cosC we get

tanB tanC = 2

HENCE PROVED

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