Evaluate the following expression ( tan^x +1) /( tan^x -1). if (sinx + cosx)/(sinx - cosx) = 5/4

Solution:

$\frac{sinx + cosx}{sinx - cosx} = \frac{5}{4}$

$4(sinx + cosx) = 5(sinx - cosx)$            [Cross Multiplication]

$4sinx + 4cosx = 5sinx - 5cosx$

$5cosx + 4cosx = 5sinx - 4sinx$              [Taking common terms on each side]

$9cosx = sinx$

$9 = \frac{sinx}{cosx}$                           [Dividing by cosx on both side]

$tanx = 9$                                                 [ As, $\frac{sinx}{cosx}= tanx$] 

Placing this value in,  $\frac{tan^x + 1}{tan^x - 1}$

$= \frac{9^2 + 1}{9^2 - 1}$

$= \frac{81 + 1}{81 - 1}$

$= \frac{82}{80}$

$= \frac{41}{40}$