If β is such that sin2 β is not equals to zero, then show that the expression[x2+2x cos2α+1] / [x2+2x cosβ +1 ] for x belonging to R always lie between cos2α / cos2β and sin2α/sin2β.

Lets take y=[x²+2x cos2α+1]/[x²+2x cos2β +1] for x belonging to R

so, its [x²+2x cos2α+1] - y*[x²+2x cos2β +1] = 0

       (1-y)x²+2x(cos2α-cos2β)+(1-y) = 0

       the equation has real roots, when discriminant os positive

i.e., (cos2α-cos2β)² - (1-y)² > 0

      [(cos2α-cos2β)+(1-y)].[(cos2α-cos2β)-(1-y)] > 0

       [ y(1+cos2β)-(1+cos2α) ].[ y(1-cos2β)-(1-cos2α) ] < 0

           (1+cos2α)/(1+cos2β)  <  y  <  (1-cos2α)/(1-cos2β)

 so, y always lie between cos²α/cos²β < y < sin²α/sin²β.

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The given expression can be written as 1 + 2(cos 2A - cos 2B)/(x + 1/x + 2 cos 2B)

Since x + 1/x >= 2 or <= -2, we have  1/(cos 2B -1) <= 1/(x+1/x + 2 cos 2B) <= 1/2(1+cos 2B)

Hence 2(cos 2A - cos 2B)/(x + 1/x + 2 cos 2B) lies between 2(cos 2A - cos 2B)/2(1+cos 2B) and 2(cos 2A - cos 2B)/2(cos 2B-1)

so that 1 + 2(cos 2A - cos 2B)/(x + 1/x + 2 cos 2B) lies between 2(1+cos 2A)/2(1+cos 2B) and 2(1-cos 2A)/2(1-cos 2B) i.e.

between cos²A/cos²B and sin²A/sin²B