2009/016) If a cos² x + b sin² x = c, express tan² x in terms of a, b and c

a cos² x + b sin² x = c
= c(sin ^2 x + cos ^2x) (knowing that 1 = sin ^2 x + cos ^ 2 x)
= c sin ^2 x + c cos ^2 x
or a cos^ 2 x - c cos^2 x = c sin ^2 x - b sin^ 2x
or (a-c) cos ^2x = (c-b) sin ^2 x

or (a-c)/(c-b) = sin ^2 x/ cos ^2 x = tan ^2 x

so tan ^2x = (a-c)/(c-b)