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
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Correct Answer: n 2
Solution :
Given that $\frac{\cos\alpha }{\cos\beta }=m$ and $\frac{\cos\alpha }{\sin\beta }=n$, we can express $\cos\: \alpha$ in terms of $m$ and $n$ as follows:
$⇒\cos\alpha = m×cos\beta =n× \sin\beta$
Squaring both sides, we get:
$⇒(m × \cos\beta)^2 = (n × \sin\beta)^2$
This simplifies to:
$⇒m^2 × \cos^2\beta = n^2 × (1 - \cos^2\beta)$
Rearranging terms, we get:
$⇒m^2 × \cos^2\beta + n^2 × \cos^2\beta = n^2$
Factoring out $cos^2\beta$, we get:
$⇒(m^2 + n^2) × \cos^2\beta = n^2$
Hence, the correct answer is n
2
.
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