Question

In any triangle, prove that
(− )÷(− ) = ÷ B​

Answer 1

Answer:

cosA+cosB+cosC=1+4sin

2

A

sin

2

B

sin

2

C

>1 ..(1)

as neither of sin

2

A

,sin

2

B

,sin

2

C

is −ive or zero.

Again cosA+cosB+cosC

=2cos

2

A+B

cos

2

A−B

+1−2sin

2

2

C

≤2sin

2

C

⋅1+1−2sin

2

2

C

∵0≤cos

2

A−B

≤1

=−2[s

2

−s−

2

1

], where s=sin

2

C

=−2[(s−

2

1

)

2

−

2

1

−

4

1

]

=

2

3

−(s−

2

1

)

2

≤

2

3

∴cosA+cosB+cosC≤3/2 .(2)

In other words, ∑cosA≯3/2