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`        In triangle ABC prove that  cosA+cosB+cosC is always positive`
7 years ago

akhil samir killawala
18 Points
```										cosA+cosB+cosC=1+4sin(a/2)sin(b/2)sinc(c/2)
.. LHS  = ( cos A + cos B ) + cos C  = { 2 · cos[ ( A+B) / 2 ] · cos [ ( A-B) / 2 ] } + cos C  = { 2 · cos [ (π/2) - (C/2) ] · cos [ (A-B) / 2 ] } + cos C  = { 2 · sin( C/2 ) · cos [ (A-B) / 2 ] } + { 1 - 2 · sin² ( C/2 ) }  = 1 + 2 sin ( C/2 )· { cos [ (A -B) / 2 ] - sin ( C/2 ) }  = 1 + 2 sin ( C/2 )· { cos [ (A-B) / 2 ] - sin [ (π/2) - ( (A+B)/2 ) ] }  = 1 + 2 sin ( C/2 )· { cos [ (A-B) / 2 ] - cos [ (A+B)/ 2 ] }   = 1 + 2 sin ( C/2 )· 2 sin ( A/2 )· sin( B/2 ) ... ... ... (2)  = 1 + 4 sin(A/2) sin(B/2) sin(C/2)  = RHS ..
from this u can proove n deduce your qustion..
```
7 years ago
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