Question

if tanx=(b)/(a), find the value of \sqrt((a+b)/(a-b))+\sqrt((a-b)/(a+b)) in terms of x

Answer 1

Let's find at first $\sqrt((a+b)/(a-b))$

Let's find (a+b)(a-b)....

we should divide both side of fraction by b and we will get the following:
$\frac{ \frac{a+b}{b} }{ \frac{a-b}{b} } \\ \frac{ \frac{a}{b} + 1 }{ \frac{a}{b} - 1 } \\ \frac{ tanx + 1 }{ tanx - 1 }$

and (a-b)/(a+b) is same as (1/[(a+b)/(a-b)]) so we can use previous answer and finally we will have (a-b)/(a+b) = 1/[ (tanx + 1) / (tanx - 1)] = (tanx - 1)/(tanx + 1) and final answer will be this:

$\sqrt{ \frac{tanx + 1}{tanx - 1} } + \sqrt{ \frac{tanx - 1}{tanx + 1} }$