Question : If $\sin \theta+\cos \theta=\sqrt{2} \cos \theta$, then find $\frac{\sin \theta-\cos \theta}{\sin \theta}$:

Option 1: $-\sqrt{2}$

Option 2: $-1$

Option 3: $1$

Option 4: $\sqrt{2}$

Question : If $\cos ^2 \theta-\sin ^2 \theta=\tan ^2 \phi$, then which of the following is true?

Option 1: $\cos \theta \cos \phi=1$

Option 2: $\cos ^2 \phi-\sin ^2 \phi=\tan ^2 \theta$

Option 3: $\cos ^2 \phi-\sin ^2 \phi=\cot ^2 \theta$

Option 4: $\cos \theta \cos \phi=\sqrt{2}$

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 $21 \tan \theta=20$, then $(1+\sin \theta+\cos \theta):(1-\sin \theta+\cos \theta)=$?

Option 1: 5 : 2

Option 2: 3 : 1

Option 3: 7 : 3

Option 4: 2 : 1