Question : If $\operatorname{cos} \theta+\operatorname{sin} \theta=\sqrt{2} \operatorname{cos} \theta$, find the value of $(\cos \theta-\operatorname{sin} \theta)$

Option 1: $\sqrt{2} \sin \theta$

Option 2: $\sqrt{2} \cos \theta$

Option 3: $\frac{1}{\sqrt{2}} \sin \theta$

Option 4: $\frac{1}{2}\cos \theta$

Question : If $\sin \theta+\cos \theta=\frac{\sqrt{11}}{3}$, then what is $\sin \theta-\cos \theta$?

Option 1: $\frac{\sqrt{7}}{4}$

Option 2: $\frac{\sqrt{7}}{3}$

Option 3: $\frac{\sqrt{5}}{3}$

Option 4: $\frac{\sqrt{5}}{2}$

Question : If $\cos \theta-\sin \theta=\sqrt{2} \sin \theta$, then $(\cos \theta+\sin \theta)$ is:

Option 1: $-\sqrt{2} \cos \theta$

Option 2: $\sqrt{2} \cos \theta$

Option 3: $\sqrt{2} \tan \theta$

Option 4: $-\sqrt{2} \sin \theta$

Question : If $\sin \theta+\cos \theta=\frac{\sqrt{11}}{3}$, then the value of $(\cos \theta-\sin \theta)$ is:

Option 1: $\frac{\sqrt{5}}{3}$

Option 2: $\frac{7}{3}$

Option 3: $\frac{5}{3}$

Option 4: $\frac{\sqrt{7}}{3}$

Question : If $\frac{\cos \theta}{1-\sin \theta}+\frac{\cos \theta}{1+\sin \theta}=4,0^{\circ}<\theta<90^{\circ}$, then what is the value of $(\sec \theta+\operatorname{cosec} \theta+\cot \theta) ?$

Option 1: $1+2 \sqrt{3}$

Option 2: $\frac{1+2 \sqrt{3}}{3}$

Option 3: $\frac{2+\sqrt{3}}{3}$

Option 4: $2+\sqrt{3}$