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

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 $\sin \theta \cos \theta=\frac{\sqrt{2}}{3}$,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 : The value of expression $4\left(\sin ^6 A+\cos ^6 A\right)-6\left(\sin ^4 A+\cos ^4 A\right)+8$ is:

Option 1: 4

Option 2: 8

Option 3: 7

Option 4: 6

Question : If $\sin A+\sin ^2 A=1$, then the value of the expression $\left(\cos ^2 A+\cos ^4 A\right)$ is

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

Option 2: $1$

Option 3: $2$

Option 4: $\frac{1}{2}$