NDA Math Trigonometric Ratios of Angles and Equations

∵ 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

=cos‌‌

C

2

We can see that statement (3) is not correct
Hence only 2nd statement is correct.