show that:tan(x+a)/tan(x-a) cannot lie between the values tan square(pie/4-a) and tan square(pie/4+a)

Dear ravi

let y  = tan(x+a) /tan(x-a)

        =tan{(x-∏/4) +(∏/4 +a)} / tan{( x-∏/4) + (∏/4-a)}

let tan(x-∏/4) =X

 tan(∏/4 +a) =c

tan(∏/4 -a) =d

now open the expression

y =[(x+c)/(1-xc)] * [(1-xd)/(x+d)]

cross multiply and simplyfy

x²(d-yc) +x(y-1)(1-cd) +(yd-c) =0

for this function have real value discrimnant must be greater than zero

  D≥0

put the discrimnant greater than zero you will get the desired result

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