Question

if m=cos α / cos β and n=cos α / sin β ,show that (m^2+n^2)cos^2β=n^2

Answer 1

answer in the attachment
hope it helps.

Answer 2

It's difficult to write alpha and beta every write. I gonna replace them with a and b.

Given m = cos a/cos b    ------------- (1)

           n = cos a/sin b      ------------ (2).


LHS = (m^2 + n^2)cos^2 b

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

Here LCM is cos^2b sin^2b

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

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

        = cos^2 a(1)/sin^2b

        = cos^2 a/sin^2 b              ------------------------ (3)


From Equation (2), Equation (3) can be written as

        = n^2.


Therefore (m^2 + n^2)cos^2b = n^2.


Hope this helps!