Question : If $\sin A-\cos A=\frac{\sqrt{3}-1}{2}$, then the value of $\sin A\cdot \cos A$ is:

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

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

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

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

Question : If $\theta$ is an acute angle and $\sin \theta \cos \theta=2 \cos ^3 \theta-\frac{1}{4} \cos \theta$, then the value of $\sin \theta$ is:

Option 1: $\frac{\sqrt{15}-1}{8}$

Option 2: $\frac{\sqrt{15}-1}{4}$

Option 3: $\frac{\sqrt{15}+1}{4}$

Option 4: $\frac{\sqrt{15}-1}{2}$

Question : If $\theta$ is an acute angle and $\sin \theta=\frac{13}{19}$, what is the value of $\cos \theta?$

Option 1: $\frac{6}{19}$

Option 2: $\frac{10 \sqrt{2}}{19}$

Option 3: $\frac{14}{19}$

Option 4: $\frac{8 \sqrt{3}}{19}$

Question : What is the value of the expression: $\sin A(1+\frac{\sin A}{\cos A})+\cos A(1+\frac{\cos A}{\sin A})$?

Option 1: $\sec A+\operatorname{cosec}A$

Option 2: $\sin \mathrm{A}+\cos \mathrm{A}$

Option 3: $\sin \mathrm{A}-\cos \mathrm{A}$

Option 4: $\sec \mathrm{A}-\operatorname{cosec} \mathrm{A}$

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: $\frac{1}{3}$