Question : If $\alpha +\beta =90^\circ$ and $\alpha :\beta =2:1,$ then the ratio of $\cos \alpha$ to $\cos \beta$ is:

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

Option 2: $1:3$

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

Option 4: $1:2$

Question : Using $\operatorname{cosec}(\alpha+\beta)=\frac{\sec \alpha \times \sec \beta \times \operatorname{cosec} \alpha \times \operatorname{cosec} \beta}{\sec \alpha \times \operatorname{cosec} \beta+\operatorname{cosec} \alpha \times \sec \beta}$, find the value of $\operatorname{cosec} 75°$.

Option 1: $\frac{\sqrt{6}+\sqrt{2}}{4}$

Option 2: $\frac{\sqrt{6}-\sqrt{2}}{4}$

Option 3: $\sqrt{6}-\sqrt{2}$

Option 4: $\sqrt{6}+\sqrt{2}$

Question : If $\tan(\alpha -\beta)=1,\sec(\alpha +\beta)=\frac{2}{\sqrt{3}}$ and $\alpha ,\beta$ are positive, then the smallest value of $\alpha$ is:

Option 1: $142\frac{1}{2}°$

Option 2: $187\frac{1}{2}°$

Option 3: $7\frac{1}{2}°$

Option 4: $37\frac{1}{2}°$

Question : If $\alpha$ and $\beta$ are the roots of equation $x^{2}-2x+4=0$, then what is the equation whose roots are $\frac{\alpha ^{3}}{\beta ^{2}}$ and $\frac{\beta ^{3}}{\alpha ^{2}}?$

Option 1: $x^{2}-4x+8=0$

Option 2: $x^{2}-32x+4=0$

Option 3: $x^{2}-2x+4=0$

Option 4: $x^{2}-16x+4=0$

Question : If $\tan \alpha = 6$, then $\sec \alpha$ is equal to:

Option 1: $\sqrt{7}$

Option 2: $\sqrt{5}$

Option 3: $\sqrt{37}$

Option 4: $\sqrt{35}$