# Thread: Find tan A/2 + tan C/2?

1. ## Find tan A/2 + tan C/2?

In a triangle ABC, if sides a,b and c are in A.P(Arithmetic Progression), angle B = pi/4, find tan A/2 + tan C/2?

2. Originally Posted by fardeen_gen
In a triangle ABC, if sides a,b and c are in A.P(Arithmetic Progression), angle B = pi/4, find tan A/2 + tan C/2?
This is how I would start off.
$\displaystyle \tan \frac{A}{2} + \tan \frac{C}{2} = \frac{\sin \frac{A}{2}}{\cos \frac{A}{2}} + \frac{\sin \frac{C}{2}}{\cos \frac{C}{2}} = \frac{\sin \left( \frac{A}{2} + \frac{C}{2} \right)}{\cos \frac{A}{2}\cos \frac{C}{2}}$

But $\displaystyle A+B+C = \pi$ and $\displaystyle B=\frac{\pi}{4}$ this means $\displaystyle \sin \left( \frac{A}{2}+\frac{C}{2} \right) = \sin \frac{3\pi}{8}$.

Thus, the problem comes down to computing $\displaystyle \cos \frac{A}{2}\cos \frac{C}{2}$.
Which looks easier.

3. If A + C = 3pi/4, sin(A + C) = sin 3pi/4. But does this necessarily imply by some formula(which I seem to be forgetting) that sin ((A + C)/2) = sin 3pi/8? Please help.

4. Originally Posted by fardeen_gen
If A + C = 3pi/4, sin(A + C) = sin 3pi/4. But does this necessarily imply by some formula(which I seem to be forgetting) that sin ((A + C)/2) = sin 3pi/8? Please help.
Hey you !

A+C=3pi/4
Then (A+C)/2=3pi/8
Thus sin((A+C)/2)=sin(3pi/8)

But for your arithmetic progression, no idea !!

5. Ouch!!It seems that trying to delve into something deep always makes me forget the most basic and fundamental things concerned.

6. I solved it! Use conditional identity tan A/2.tan B/2 + tan B/2.tan C/2 + tan C/2. tan A/2 = 1, for A + B + C = pi. That made it easy