(sin(θ/2)+cos(θ/2))^2=?

🤔 প্রশ্ন: \((\sin(\frac{\theta}{2}) + \cos(\frac{\theta}{2}))^2 = ? \)

💡 সমাধান:

আমরা জানি, \((a + b)^2 = a^2 + 2ab + b^2\)

সুতরাং, \((\sin(\frac{\theta}{2}) + \cos(\frac{\theta}{2}))^2 = \sin^2(\frac{\theta}{2}) + 2\sin(\frac{\theta}{2})\cos(\frac{\theta}{2}) + \cos^2(\frac{\theta}{2})\)

আমরা আরও জানি, \( \sin^2(x) + \cos^2(x) = 1\) এবং \(2\sin(x)\cos(x) = \sin(2x)\)

তাহলে, \( \sin^2(\frac{\theta}{2}) + \cos^2(\frac{\theta}{2}) = 1\) এবং \(2\sin(\frac{\theta}{2})\cos(\frac{\theta}{2}) = \sin(2 \cdot \frac{\theta}{2}) = \sin(\theta)\)

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

✅ উত্তর: \(1 + \sin(\theta)\)