Question : If $\frac{\cos \alpha}{\sin \beta} = 10$ and $\frac{\cos \alpha}{\cos \beta} = 11$, the value of $\cos ^2 \beta$ is:

Option 1: $\frac{121}{132}$

Option 2: $\frac{100}{221}$

Option 3: $\frac{88}{108}$

Option 4: $\frac{221}{121}$

Question : If $\alpha+\beta=90^{\circ}$ and $\alpha=2 \beta$, then the value of $3 \cos ^2 \alpha-2 \sin ^2 \beta$ is equal to:

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

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

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

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

Question : Simplify the following.
$\frac{\sin^3 \alpha+\cos^3 \alpha}{\sin \alpha+\cos \alpha}$

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

Option 2: $\tan \alpha$

Option 3: $1-\sin \alpha \cos \alpha$

Option 4: $\sec \alpha$

Question : If $\frac{\cos\alpha}{\cos\beta}= m$ and $\frac{\cos\alpha}{\sin\beta}= n$, then the value of $(m^{2}+n^{2})\cos^{2}\beta$ is:

Option 1: n²

Option 2: m²

Option 3: mn

Option 4: 1

Question : If $\frac{\cos \beta}{\sec \alpha}=15$ and $\frac{\sin \beta}{\sec \alpha}=16$, the value of $\sin ^2 \beta$ is:

Option 1: $\frac{256}{481}$

Option 2: $-\frac{256}{481}$

Option 3: $\frac{481}{256}$

Option 4: $-\frac{481}{256}$