Question : Which of the following options gives an expression equivalent to $\sin \ (A + B)$?

Option 1: $\cos\ A \cos\ B - \sin \ A \sin \ B$

Option 2: $\sin\  A \cos \ B + \cos\ A\ \sin \ B$

Option 3: $\cos \ A \cos\ B+\sin\ A \sin \ B$

Option 4: $\sin \ A \cos \ B - \cos \ A \sin \ B$

Question : Solve the following to find its value in terms of trigonometric ratios.
$(\sin A + \cos A)(1 - \sin A \cos A)$

Option 1: $\sin^3A+\cos^3A$

Option 2: $\sin^2A-\cos^2A$

Option 3: ${[\cos A-\sin A]\left[\sin ^2 A+\cos ^2 A\right]}$

Option 4: $\sin^3A-\cos^3A$

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 : What is the value of the expression $\cos 2 A \cos 2 B+\sin ^2(A-B)-\sin ^2(A+B)$?

Option 1: $\sin (2 A-2 B)$

Option 2: $\sin (2 A+2 B)$

Option 3: $\cos (2 A+2 B)$

Option 4: $\cos (2 A-2 B)$

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: $\sin ^2 \theta-\cos ^2 \theta$