If $ \frac{\cos \alpha}{\cos \beta}=m, \quad \frac{\cos \alpha}{\sin

1.	If $ \frac{\cos \alpha}{\cos \beta}=m, \quad \frac{\cos \alpha}{\sin \beta}=n $ show that $ \left(m^{2}+n^{2}\right) \cos ^{2} \beta=n^{2} $
Answer» L.H.S= (m^2+n^2)cos^2b = {(cosa/cosb)^2+(cosa/sinb)^2}cos^2b = {(cos^2a/cos^2b) +(cos^2a/sin^2b)}cos^2b = (cos^2a.sin^2b +cos^2a.cos^2b / cos^2b.sin^2b)cos^2b = cos^2a(sin^2b+cos^2b) / sin^2b = cos^2a / sin^2b R.H.S= n^2 =(cosa/sinb)^2 = cos^2a / sin^2b

If $ \frac{\cos \alpha}{\cos \beta}=m, \quad \frac{\cos \alpha}{\sin \beta}=n $ show that $ \left(m^{2}+n^{2}\right) \cos ^{2} \beta=n^{2} $

Answer»

L.H.S= (m^2+n^2)cos^2b

= {(cosa/cosb)^2+(cosa/sinb)^2}cos^2b

= {(cos^2a/cos^2b) +(cos^2a/sin^2b)}cos^2b

= (cos^2a.sin^2b +cos^2a.cos^2b / cos^2b.sin^2b)cos^2b

= cos^2a(sin^2b+cos^2b) / sin^2b

= cos^2a / sin^2b

R.H.S= n^2

=(cosa/sinb)^2

= cos^2a / sin^2b