tanθ / (1 + tan²θ) =?

প্রশ্ন: \( \frac{\tan{\theta}}{1 + \tan^2{\theta}} = ? \) 🤔

আমরা জানি, \( 1 + \tan^2{\theta} = \sec^2{\theta} \)। 🤓

সুতরাং, \( \frac{\tan{\theta}}{1 + \tan^2{\theta}} = \frac{\tan{\theta}}{\sec^2{\theta}} \) 😊

আবার, \( \tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} \) এবং \( \sec{\theta} = \frac{1}{\cos{\theta}} \)। 🤩

তাহলে, \( \frac{\tan{\theta}}{\sec^2{\theta}} = \frac{\frac{\sin{\theta}}{\cos{\theta}}}{\frac{1}{\cos^2{\theta}}} \) 👍

\( = \frac{\sin{\theta}}{\cos{\theta}} \cdot \cos^2{\theta} \) 🎉

\( = \sin{\theta} \cos{\theta} \) 🥳

অতএব, \( \frac{\tan{\theta}}{1 + \tan^2{\theta}} = \sin{\theta} \cos{\theta} \)। ✅