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 $\sin(A+B)=\sin A\cos B+\cos A \sin B$, then the value of $\sin75°$ is:

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

Option 2:

$\frac{\sqrt{2}+1}{2\sqrt{2}}$

Option 3:

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

Option 4:

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

Question : If $\sin m+\sin n=p, \cos m+\cos n=q$, then find the value of $\sin m \times \sin n+\cos m \times \cos n$.

Option 1: $p^2+q^2-2$

Option 2: $\frac{p^2+q^2-2}{2}$

Option 3: $p+q-p q$

Option 4: $p+q+p q$

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 $\sin \alpha=\frac12$ and $\sin \beta=\frac12$, then what is the value of $\cos (\alpha+\beta)$? $(0°<\alpha, \beta<90° )$

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

Option 2: $\frac{1}{ 4}$

Option 3: $\frac12$

Option 4: $1$