Question

if A , B and ,C are angles ina triangle then tan (A+B by 2 ) tan( C by 2)+ tan (B+C by 2) tan( A by 2) + tan ( C+A by 2 ) tan ( B by 2 ) =​

Answer 1

Answer:

A+B+C=180

o

C=180

o

−(A+B)⇒

2

C

=

2

π

−

2

A+B

tan(C/2)=tan(π/2−(A+B)/2)=

tan((A+B)/2)

1

=

tan(A/2)+tan(B/2)

1−tan(A/2)tan(B/2)

Let a=tan(A/2),b=tan(B/2),c=tan(C/2) all positive, the constraint becomes

c=(1−ab)/(a+b)

which is equivalent to ab+bc+ca=1

and we know that,

a

2

+b

2

+c

2

≥ab+bc+ca⇒a

2

+b

2

+c

2

≥1⇒tan

2

(A/2)+tan

2

(B/2)+tan

2

(C/2)≥1

Therefore, Answer is ≥1