Correct Answer: $\frac{100}{221}$

Solution : Given: $\frac{\cos \alpha}{\sin \beta} = 10$ and $\frac{\cos \alpha}{\cos \beta} = 11$
⇒ ${\cos \alpha} = 10\  {\sin \beta}$ and ${\cos \alpha} = 11\ {\cos \beta}$
⇒ $10\ {\sin \beta} = 11\ {\cos \beta}$
⇒ $\frac{\sin \beta}{\cos \beta} = \frac{11}{10}$
⇒ ${\tan \beta} = \frac{11}{10}$
We know, $\sec^2 \beta-\tan^2\beta=1$
⇒ $\sec^2 \beta-(\frac{11}{10})^2=1$
⇒ $\sec^2 \beta=1+\frac{121}{100}$
⇒ $\sec^2 \beta=\frac{221}{100}$
$\therefore {\cos^2\beta} = \frac{100}{221}$
Hence, the correct answer is $\frac{100}{221}$.