যদি  tanalpha-tanbeta=p, cotbeta-cotalpha=q, alpha-beta=theta  হয়,  তবে  cottheta  এর মান কত ?  

Explanation:

Another Explanation (5): দেওয়া আছে, \[ \tan \alpha - \tan \beta = p \] \[ \cot \beta - \cot \alpha = q \] \[ \alpha - \beta = \theta \] আমাদের \(\cot \theta\) এর মান নির্ণয় করতে হবে। আমরা জানি, \(\cot \theta = \frac{1}{\tan \theta}\) তাহলে, \(\tan \theta = \tan (\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta} = \frac{p}{1 + \tan \alpha \tan \beta}\) আবার, \(q = \cot \beta - \cot \alpha = \frac{1}{\tan \beta} - \frac{1}{\tan \alpha} = \frac{\tan \alpha - \tan \beta}{\tan \alpha \tan \beta} = \frac{p}{\tan \alpha \tan \beta}\) সুতরাং, \(\tan \alpha \tan \beta = \frac{p}{q}\) এখন, \(\tan \theta = \frac{p}{1 + \frac{p}{q}} = \frac{p}{\frac{q+p}{q}} = \frac{pq}{p+q}\) অতএব, \(\cot \theta = \frac{1}{\tan \theta} = \frac{p+q}{pq} = \frac{p}{pq} + \frac{q}{pq} = \frac{1}{q} + \frac{1}{p}\) সুতরাং, \(\cot \theta = \frac{1}{p} + \frac{1}{q}\) 🥳