Question : If $\cot A=1, \sin B=\frac{1}{\sqrt{2}}$, find the value of $\sin (A+B)-\cot (A+B)$.

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

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

Option 3: $0$

Option 4: $1$

Question : Find the value of $\frac{\cos^2 15^{\circ}-\sin^2 15^{\circ}}{\cos^2 145^{\circ}+\sin^2 145^{\circ}}$.

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

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

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

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

Question : In a right triangle for an acute angle $x$, if $\sin x=\frac{3}{7}$, then find the value of $\cos x$.

Option 1: $\frac{2}{7}$

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

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

Option 4: $\frac{2\sqrt{10}}{7}$

Question : If $\tan A=\frac{2}{3}$, then find $\sin A$.

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

Option 2: $\frac{2}{\sqrt{13}}$

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

Option 4: $\frac{3}{\sqrt{13}}$

Question : If $\sin \theta+\cos \theta=\frac{\sqrt{3}-1}{2 \sqrt{2}}$, then what is the value of $\tan \theta+\cot \theta$?

Option 1: $8(\sqrt{3}-2)$

Option 2: $12(\sqrt{3}-2)$

Option 3: $12(\sqrt{3}+2)$

Option 4: $8(\sqrt{3}+2)$