Question : What is the value of $\frac{\sin \theta+\cos \theta}{\sin \theta-\cos \theta}+\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta}$?

Option 1: $\frac{1}{\left(\sin ^2 \theta-\cos ^2 \theta\right)}$

Option 2: $2\left(\sin ^2 \theta-\cos ^2 \theta\right)$

Option 3: $\frac{2}{\left(\sin ^2 \theta-\cos ^2 \theta\right)}$

Option 4: $\sin ^2 \theta-\cos ^2 \theta$

Question : If $\sin \theta \cos \theta=\frac{1}{\sqrt{3}}$ then the value of $\left(\sin ^4 \theta+\cos ^4 \theta\right)$ is:

Option 1: $1$

Option 2: $\frac{5}{3}$

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

Option 4: $\frac{1}{3}$

Question : If $\sin \theta \cos \theta=\frac{\sqrt{2}}{3}$,then the value of $\left(\sin ^6 \theta+\cos ^6 \theta\right)$ is:

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

Option 2: $\frac{4}{3}$

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

Option 4: $\frac{5}{3}$

Question : If $x=a\left ( \sin\theta+\cos\theta \right )$ and $y=b\left ( \sin\theta-\cos\theta \right )$, then the value of $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}$ is:

Option 1: 4

Option 2: 3

Option 3: 1

Option 4: 2

Question : Simplify the given equation: $(1+\tan ^2 A)(1+\cot ^2 A)=?$

Option 1: $\frac{1}{\cos ^2 A\left(1+\sin ^2 A\right)}$

Option 2: $\frac{1}{\sin ^2 A\left(1-\sin ^2 A\right)}$

Option 3: $\frac{1}{\sin ^2 A+\operatorname{cosec}^2 A}$

Option 4: $\frac{1}{\sin ^2 A\left(1+\cos ^2 A\right)}$