How to prove that $\tan^2(\frac\theta2)= \tan^2(\frac\alpha2)\tan^2(\frac\beta2)$?

Question

I'm unable to solve this question:
$\cos(\theta)=\dfrac{\cos(\alpha)+\cos(\beta)}{1+\cos(\alpha) \cos(\beta)}$

Prove: $\tan^2\left(\frac\theta2\right)= \tan^2\left(\frac\alpha2\right)\tan^2\left(\frac\beta2\right)$

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I have tried the following:

1. Using the indentity: $\cos(\theta)=\dfrac{1-\tan^2\left(\frac\theta2\right)}{1+\tan^2\left(\frac\theta2\right)}$
   
2. Diving by $\cos (\alpha) \cos (\beta)$
   
3. Creating a triangle, to find $\tan(\theta)= \sin \alpha \sin \beta$
   

Every time I got a huge complex equation with roots. Any help would be appreciated.

Answer 1

You may try componendo and dividendo :

$\dfrac{1-\tan^2\left(\frac\theta2\right)}{1+\tan^2\left(\frac\theta2\right)}=\dfrac{\cos(\alpha)+cos(\beta)}{1+\cos(\alpha) \cos(\beta)} \\ \iff \dfrac{-2\tan^2\left(\frac\theta2\right)}{2} \stackrel{\color{red}{*}}{=} \dfrac{\cos(\alpha)+cos(\beta) - 1-\cos(\alpha)\cos(\beta)}{\cos(\alpha)+cos(\beta) + 1+\cos(\alpha)\cos(\beta)} \\ \iff \tan^2\left(\frac\theta2\right) = \dfrac{(1-\cos(\alpha))(1-\cos(\beta))}{(1+\cos(\alpha))(1+\cos(\beta))} $

$\color{Red}{*} :$ componendo and dividendo

Answer 2

$$ \tan ^2 {\frac {\theta}{2}} = \frac{\sin^2 {\frac {\theta}{2}} }{\cos^2 {\frac {\theta}{2}}} = \frac{ \left({\dfrac{1 - \cos \theta}{2}}\right)}{\left({\dfrac{1 + \cos \theta}{2}}\right)} \;\; \text{(since $\cos 2 \theta = \cos^2 \theta - \sin^2 \theta= 1 - 2 \sin^2 \theta = 2 \cos^2\theta - 1$ )}$$

Now substituting for $\cos \theta$ and factorising we have,

$$ = \frac{1 + \cos \alpha \cos \beta - \cos \alpha - \cos \beta }{ 1 + \cos \alpha \cos \beta + \cos \alpha + \cos \beta} = \frac{(1 - \cos \alpha)(1 - \cos \beta)}{(1 + \cos \alpha)(1 + \cos \beta)} $$

Now use the same formulas for $cos 2 \theta $ mentioned above strategically to negate the $1$ present in all the factors and to introduce $\dfrac {\alpha}{2} $. Then,

$$ = \frac{[1 - (1 - 2 \sin^2 \frac {\alpha}{2} ) ][1 - ( 1 - 2 \sin^2 \frac {\beta}{2})]}{[1 + (2 \cos^2 \frac {\alpha}{2} - 1)][1 + (2 \cos ^2 \frac {\beta}{2} - 1)]} $$

Remember $\tan \theta = \dfrac{\sin \theta}{\cos \theta}$.

Now Simplify!!