1.Prove that for real x, tan(x+a)/tan(x-a) when x≠∏/4, 3∏/4 cannot lie between (1-sin2a)/(1+sin2a) and (1+sin2a)/(1-sin2a)

2.Find the set of possible values of tan(x+∏/6)/tanx where x is any real angle.

Dear manoj

let

    y = tan(x+a) /tan(x-a)

       = sin(x+a)cos(x-a) / cos(x+a) sin(x-a)

       =[sin2x + sin2a]/[sin2x -sin2a]

 use componendo and divedendo

(y+1)/(y-1) = sin2x /sin2a

or

   sin2x = (y+1)sin2a /(y-1)

for every real value of x

-1≤  sin2x ≤1

 so  -1≤  (y+1)sin2a /(y-1) ≤1

 simplyfy each side of inequality

  (1-sin2a)/(1+sin2a)  ≤  y  ≤   (1-sin2a)/(1+sin2a)

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