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: $\frac{221}{121}$
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 $\frac{100}{221}$.
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