∵ ABC is an equilateral triangle ∴A=B=C=60° L.H.S=3tan(A+B)tanC =3tan120∘tan60∘ =3(−√3)(√3) =−9≠1 Hence statement (1) is incorrect. Statement-2 ABC is a triangle such that A =78° and B = 66° C=180−(78+66)=180−144=36∘
A
2
+C=
78
2
+36∘ =39+36=75∘ tan(
A
2
+C)<tanA ⇒tan75∘<tan78∘ Hence statement (2) is correct. Statement (3) In a triangleABC A+B+C=180∘ A+B=180°−C
A+B
2
=
180∘−C
2
⇒
A+B
2
=90−
C
2
⇒tan(
A+B
2
)=tan(90−
C
2
) ⇒tan(
A+B
2
)=cot
C
2
.....(1)
∴tan(
A+B
2
)⋅sin
C
2
=cot
C
2
⋅sin
C
2
=cos
C
2
⇒tan(
A+B
2
)⋅sin
C
2
=cos
C
2
We can see that statement (3) is not correct Hence only 2nd statement is correct.