Proving $\frac{s^2}{c}\left[\tan\frac\alpha2+\tan\frac\beta2\right]\left[\tan\frac\alpha2\tan\frac\beta2\right]=(s−c)\cot\frac\gamma2$ in any triangle

Question

The question below was a part for my board exam of fbise, the key provided by the board that this statement would be proved through law of cosine, law tangent or law of sine, im confused furthermore I have searched this question over the internet and I haven't found something related to it, so any help would be recommended.

In triangle $ABC$ (with usual notations) such that $$ \alpha+\beta+\gamma = \pi $$ then prove that, $$ \frac{s^2}{c}\left[\tan\left(\frac{\alpha}{2}\right)+ \tan \left(\frac{\beta}{2}\right)\right]\left[\tan \left(\frac{\alpha}{2}\right) \tan \left(\frac{\beta}{2}\right)\right] = (s − c) \cot \left(\frac{\gamma}{2}\right) $$

where $ s=\frac{a+b+c}{2} $ and $a$, $b$ and $c$ are the sides of $\triangle$. Furthermore, $\alpha $, $\beta$ and $\gamma$ are the angles of triangles facing each side $a$, $b$, and $c$ respectively.

can you show $\tan(\alpha/2) + \tan(\beta/2) = \frac{c}{s}$? — Yimin, Mar 08 '24 at 16:24
@Yimin this isn't true if you consider an equilateral triangle of side some measurement and angles of 60 we couldn't get to your assumption $\frac{c}{s}$ — Raffay Elahi, Mar 08 '24 at 16:42
@JeanMarie fbise is just like an examination council which takes exam in Pakistan. — Raffay Elahi, Mar 08 '24 at 16:44
@JeanMarie Absolutely agreed! I'm just new to this community an personally my experience wasn't great, it literally took me an hour to figure out how to write notation here lol — Raffay Elahi, Mar 08 '24 at 16:46

score 1 · Accepted Answer · answered Mar 08 '24 at 17:49

1

Using $\tan(\frac{\alpha}{2})$ = $\sqrt\frac{(s-b)(s-c)}{s(s-a)}$ & $\tan(\frac{\beta}{2})$ = $\sqrt\frac{(s-c)(s-a)}{s(s-b)},$

$\&$ putting $\tan(\frac{\alpha}{2})+ \tan(\frac{\beta}{2}) = \cot(\frac{\gamma}{2})\left(1-\tan(\frac{\alpha}{2})\tan(\frac{\beta}{2})\right),$

simplifying the LHS, we get the result easily.

answered Mar 08 '24 at 17:49

Akshat Mishra

354

Could you give a reference for the proof of the first formulas ? – Jean Marie Mar 08 '24 at 22:27
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Hint : write out $sin^2(\frac{A}{2})$ = $\frac{1-cos(A)}{2}$ & use cos(A) = $\frac{b^2+c^2-a^2}{2bc}$ to get formula for $sin(\frac{A}{2})$. Proceed Similarly for the rest. – Akshat Mishra Mar 09 '24 at 01:06
Thanks for taking the time to give the outlines of the proof. – Jean Marie Mar 09 '24 at 07:42

Proving $\frac{s^2}{c}\left[\tan\frac\alpha2+\tan\frac\beta2\right]\left[\tan\frac\alpha2\tan\frac\beta2\right]=(s−c)\cot\frac\gamma2$ in any triangle

1 Answers1