Question : If $\operatorname{cosec} A+\cot A=a \sqrt{b}$, then find the value of $\frac{\left(a^2 b-1\right)}{\left(a^2 b+1\right)}$.

Option 1: $\cos A$

Option 2: $\tan A$

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

Option 4: $\frac{1}{\cot A}$

Question : If $\sec A=\frac{9}{4}$, then what is the value of $\cot A$?

Option 1: $\frac{4}{\sqrt{65}}$

Option 2: $\frac{9}{\sqrt{65}}$

Option 3: $\frac{\sqrt{65}}{9}$

Option 4: $\frac{\sqrt{65}}{4}$

Question : If cos A = $\frac{5}{13}$, then what is the value of (tan A + cot A)?

Option 1: $\frac{60}{169}$

Option 2: $\frac{109}{169}$

Option 3: $\frac{169}{60}$

Option 4: $\frac{169}{109}$

Question : If $\operatorname{cos} \theta=\frac{4}{5}$, find the value of $\operatorname{cot} \theta+\tan \theta$.

Option 1: $\frac{12}{25}$

Option 2: $\frac{25}{12}$

Option 3: $\frac{27}{12}$

Option 4: $\frac{12}{27}$

Question : If $\mathrm{A}=\cot 30^{\circ} \tan 60^{\circ}+\cot 60^{\circ} \tan 30^{\circ}$, then what is the value of A?

Option 1: $\frac{1}{ \sqrt{ 3}}$

Option 2: $\frac{10}{3}$

Option 3: $\frac{10}{ \sqrt{3}}$

Option 4: $\frac{1}{3}$