Question : If $\sec\theta+\tan\theta=m\left (> 1 \right)$, then the value of $\sin\theta$ is $\left (0^{\circ} < \theta<90^{\circ} \right)$:

Option 1: $\frac{1-m^{2}}{1+m^{2}}$

Option 2: $\frac{m^{2}-1}{m^{2}+1}$

Option 3: $\frac{m^{2}+1}{m^{2}-1}$

Option 4: $\frac{1+m^{2}}{1-m^{2}}$

Question : $\left(\frac{\tan ^3 \theta}{\sec ^2 \theta}+\frac{\cot ^3 \theta}{\operatorname{cosec}^2 \theta}+2 \sin \theta \cos \theta\right) \div\left(1+\operatorname{cosec}^2 \theta+\tan ^2 \theta\right), 0^{\circ}<\theta<90^{\circ}$, is equal to:

Option 1: $\operatorname{cosec} \theta \sec \theta$

Option 2: $\operatorname{cosec} \theta$

Option 3: $\sin \theta \cos \theta$

Option 4: $\sec \theta$

Question : If $1+2 \tan ^2 \theta+2 \sin \theta \sec ^2 \theta=\frac{a}{b}, 0^{\circ}<\theta<90^{\circ}$, then $\frac{a+b}{a-b}=?$

Option 1: $\sin \theta$

Option 2: $\cos \theta$

Option 3: $\operatorname{cosec} \theta$

Option 4: $\sec \theta$

Question : If $\cos \left(2 \theta+54^{\circ}\right)=\sin \theta, 0^{\circ}<\left(2 \theta+54^{\circ}\right)<90^{\circ}$, then what is the value of $\frac{1}{\tan 5 \theta+\operatorname{cosec} \frac{5 \theta}{2}}$?

Option 1: $3\sqrt2$

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

Option 3: $2\sqrt3$

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

Question : Let $0^{\circ}<\theta<90^{\circ}$, $\left(1+\cot ^2 \theta\right)\left(1+\tan ^2 \theta\right) × (\sin \theta-\operatorname{cosec} \theta)(\cos \theta-\sec \theta)$ is equal to:

Option 1: $ \sin \theta+\cos \theta$

Option 2: $\sin \theta \cos \theta$

Option 3: $\sec \theta \operatorname{cosec} \theta$

Option 4: $\sec \theta+\operatorname{cosec} \theta$

Question : If $3 \tan \theta=2 \sqrt{3} \sin \theta, 0^{\circ}<\theta<90^{\circ}$, then the value of $\frac{\operatorname{cosec}^2 2 \theta+\cot ^2 2 \theta}{\sin ^2 \theta+\tan ^2 2 \theta}$ is:

Option 1: $\frac{4}{13}$

Option 2: $\frac{20}{39}$

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

Option 4: $\frac{20}{27}$