If \(\frac{secθ - tanθ}{secθ + tanθ}\) = \(\frac{3}{5}\), find (\(\frac{cosecθ + cotθ}{cosecθ - cotθ}\)).

31 + 8\(\sqrt {15}\)

\(\frac{secθ - tanθ}{secθ + tanθ}\) = \(\frac{3}{5}\)

By componendo & Dividendo concept:

\(\frac{secθ}{tanθ}\) = \(\frac{5 + 3}{5 - 3}\)

\(\frac{\frac{1}{cos}}{\frac{sin}{cos}}\) = \(\frac{8}{2}\)

\(\frac{1}{sinθ}\) = \(\frac{4}{1}\)

⇒ sinθ = \(\frac{1}{4}\)   (where 1 → P and 4 → H)

⇒ cosecθ = 4

So, Base = \(\sqrt {(4)^2 - (1)^2}\) = \(\sqrt {15}\),

⇒ cotθ=  \(\frac{\sqrt {15}}{1}\) =  \(\sqrt {15}\)

Put the values and find:

⇒ \(\frac{cosecθ + cotθ}{cosecθ - cotθ}\) = \(\frac{4 + \sqrt {15}}{4 - \sqrt {15}}\) = 31 + 8\(\sqrt {15}\)