Question : If $\cot A+\cos A=p$ and $\cot A-\cos A=q$, then which of the following relation is correct?

Option 1: $\sqrt{\frac{1}{16pq}}=p^2+q^2$

Option 2: $\frac{1}{4pq}=p^2+q^2$

Option 3: $\sqrt{16pq}=p^2-q^2$

Option 4: $4pq=p^2+q^2$

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 $\sec \theta+\tan \theta=\frac{1}{\sqrt{3}}$, then the positive value of $\cot \theta+\cos \theta$ is:

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

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

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

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

Question : If $\cos\theta=\frac{p}{\sqrt{p^{2}+q^{2}}}$, then the value of $\tan\theta$ is:

Option 1: $\frac{q}{\sqrt{p^{2}-q^{2}}}$

Option 2: $\frac{q}{p}$

Option 3: $\frac{p}{p^{2}+q^{2}}$

Option 4: $\frac{q}{\sqrt{p^{2}+q^{2}}}$

Question : If $\sin\theta+\cos\theta=\sqrt{2}\cos\theta$, then the value of $\cot\theta$ is:

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

Option 2: $\sqrt{2}-1$

Option 3: $\sqrt{3}-1$

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

Question : If $a =\cot A+\cos A$ and $b =\cot A-\cos A$, then find the value of $a^2-b^2-4 \sqrt{ab}$.

Option 1: 0

Option 2: –1

Option 3: 1

Option 4: –4