Question : Which of the following is equal to $[\frac{\tan \theta+\sec \theta–1}{\tan \theta–\sec \theta+1}]$?

Option 1: $\frac{1+\sin \theta}{\cos \theta}$

Option 2: $\frac{1+\tan \theta}{\cot \theta}$

Option 3: $\frac{1+\cot \theta}{\tan \theta}$

Option 4: $\frac{1+\cos \theta}{\sin \theta}$

Question : The expression $\frac{(1-\sin \theta+\cos \theta)^2(1-\cos \theta) \sec ^3 \theta\; {\operatorname{cosec}}^2 \theta}{(\sec \theta-\tan \theta)(\tan \theta+\cot \theta)}, 0^{\circ}<\theta<90^{\circ}$, is equal to:

Option 1: $\sin \theta$

Option 2: $2 \cos \theta$

Option 3: $\cot \theta$

Option 4: $2 \tan \theta$

Question : $\frac{1+\sin \theta}{\cos \theta}$ is equal to which of the following (where $\left.\theta \neq \frac{\pi}{2}\right)?$

Option 1: $\frac{1+\cos \theta}{\sin \theta}$

Option 2: $\frac{\tan \theta+1}{\tan \theta-1}$

Option 3: $\frac{\tan \theta-1}{\tan \theta+1}$

Option 4: $\frac{\cos \theta}{1-\sin \theta}$

Question : What is $\tan \frac{\theta}{2}$?

Option 1: $\frac{\cos \theta}{1-\sin \theta}$

Option 2: $\frac{\sin \theta}{1-\cos \theta}$

Option 3: $\frac{\cos \theta}{1-\cos \theta}$

Option 4: $\frac{\sin \theta}{1+\cos \theta}$

Question : Which of the following is equal to $[\frac{\cos \theta}{\sin \theta}+\frac{\sin \theta}{\cos \theta}]$?

Option 1: $\operatorname{cosec} \theta \sec \theta$

Option 2: $\sec \theta\tan \theta$

Option 3: $\operatorname{cosec} \theta\tan \theta$

Option 4: $\cot \theta \sec \theta$