BITSAT 2024 Shift 1 Paper

When A,B, and C are the angles of a triangle, we have A+B+C=π. Thus, the following identity holds:
tan‌

A

2

‌tan‌

B

2

+tan‌

B

2

‌tan‌

C

2

+tan‌

C

2

‌tan‌

A

2

=1
Given that tan‌

A

2

=‌

1

3

and tan‌

B

2

=‌

2

3

, let tan‌

C

2

=x.
Substitute these values into the equation:
‌

1

3

×‌

2

3

+‌

2

3

×x+x×‌

1

3

=1
Simplifying, we have:
‌

2

9

+‌

2

3

x+‌

x

3

=1
Combine the x terms:
‌‌

2

9

+‌

3x

3

=1
‌‌

2

9

+x=1
Solving for x :
x=1−‌

2

9

=‌

9

9

−‌

2

9

=‌

7

9

Therefore, tan‌

C

2

=‌

7

9

.