Cot ɑ+Cot β=a এবং Tan ɑ+Tan β=b এবং ɑ+β=θ হলে cot θ=কত?

1/b - 1/a

দেওয়া আছে,

\(\cot \alpha + \cot \beta = a\)

এবং \(\tan \alpha + \tan \beta = b\)

আরও দেওয়া আছে, \(\alpha + \beta = \theta\)

আমাদের \(\cot \theta\) এর মান বের করতে হবে।

আমরা জানি, \(\tan \alpha = \frac{1}{\cot \alpha}\) এবং \(\tan \beta = \frac{1}{\cot \beta}\)

সুতরাং, \(\frac{1}{\cot \alpha} + \frac{1}{\cot \beta} = b\)

বা, \(\frac{\cot \beta + \cot \alpha}{\cot \alpha \cot \beta} = b\)

বা, \(\frac{a}{\cot \alpha \cot \beta} = b\)

সুতরাং, \(\cot \alpha \cot \beta = \frac{a}{b}\) .....(1)

এখন, \(\theta = \alpha + \beta\)

সুতরাং, \(\cot \theta = \cot (\alpha + \beta)\)

আমরা জানি, \(\cot (\alpha + \beta) = \frac{\cot \alpha \cot \beta - 1}{\cot \alpha + \cot \beta}\)

অতএব, \(\cot \theta = \frac{\frac{a}{b} - 1}{a}\)

=\(\frac{\frac{a-b}{b}}{a}\)

=\(\frac{a-b}{ab}\)

=\(\frac{a}{ab} - \frac{b}{ab}\)

=\(\frac{1}{b} - \frac{1}{a}\)

অতএব, \(\cot \theta = \frac{1}{b} - \frac{1}{a}\) 🥳