Saved Bookmarks
| 1. |
\( \left(\cot \frac{\theta}{2}-\tan \frac{\theta}{2}\right)^{2}(1-2 \tan \theta \cot 2 \theta) \) का हल होगा- (A) 1 (B) 2 (C) 3 (D) 4 |
|
Answer» The correct option is (D) 4. (cot θ/2 - tan θ/2)2 (1 - 2 tan θ cot 2θ) \(=(\cfrac{cos\,\theta/2}{sin\,\theta/2}-\cfrac{sin\,\theta/2}{cos\,\theta/2})^2\) (1 - 2 tan θ cot 2θ) \(=(\cfrac{cos^2\,\theta/2-sin^2\,\theta/2}{sin\,\theta/2\;cos\,\theta/2})^2\) (1 - 2 tan θ cot 2θ) \(=\left(\cfrac{2(cos^2\,\theta/2-sin^2\,\theta/2)}{2\,sin\,\theta/2\;cos\,\theta/2}\right)^2\) (1 - 2 tan θ cot 2θ) \(=4(\frac{cos\,\theta}{sin\,\theta})^2\) (1 - 2 tan θ cot 2θ) (∵ cos2 θ/2 - sin2 θ/2 = cos θ & 2 sin θ/2 cos θ/2 = sin θ) = 4 cot2θ \((1-\frac{2\,tan\,\theta}{tan\,2\theta})\) = 4 cot2θ \(\left(1-\frac{2\,tan\,\theta}{\frac{2\,tan\,\theta}{1-tan^2\theta}}\right)\) = 4 cot2θ (1 - (1 - tan2θ)) = 4 cot2θ (1 - 1 + tan2θ) = 4 cot2θ tan2θ = 4. |
|