Question : The value of the expression $\left[\operatorname{cot} 1^{\circ} \cdot \operatorname{cot} 2^{\circ} \cdot \operatorname{cot} 3^{\circ} \cdot \operatorname{cot} 4^{\circ} \cdot \operatorname{cot} 5^{\circ} \ldots . \operatorname{cot} 178^{\circ} \cdot \operatorname{cot} 179^{\circ}\right]$ is:

Option 1: $1235$

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

Option 3: $1$

Option 4: $0$

Question : Simplify the given equation: $(1+\tan ^2 A)(1+\cot ^2 A)=?$

Option 1: $\frac{1}{\cos ^2 A\left(1+\sin ^2 A\right)}$

Option 2: $\frac{1}{\sin ^2 A\left(1-\sin ^2 A\right)}$

Option 3: $\frac{1}{\sin ^2 A+\operatorname{cosec}^2 A}$

Option 4: $\frac{1}{\sin ^2 A\left(1+\cos ^2 A\right)}$

Question : Simplify the given expression: $\frac{\sin^2 32^{\circ}+\sin^2 58^{\circ}}{\cos^2 32^{\circ}+\cos^2 58^{\circ}}+\sin^2 53^{\circ}+\cos 53^{\circ} \sin 37^{\circ}$

Option 1: 2

Option 2: –1

Option 3: –2

Option 4: 1

Question : If $\sin\phi=\frac{5}{6}$, the value of $\cot\phi \cdot \sin\phi \cdot \cos\phi$ is:

Option 1: $\frac{6}{5}$

Option 2: $\frac{25}{36}$

Option 3: $\frac{5}{6}$

Option 4: $\frac{11}{36}$

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