Question

\( \left(\cot \frac{\theta}{2}-\tan \frac{\theta}{2}\right)^{2}(1-2 \tan \theta \cot 2 \theta) \) का हल होगा- 

(A) 1 

(B) 2 

(C) 3 

(D) 4

Answer 1

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.