If 5k = tan θ and $\frac{5}{k} = secθ$, then what is the value of $10(k^2-\frac{1}{k^2})$ ?

$-\frac{2}{5}$

tan θ = 5k

\(\frac{tan θ }{5}\) = k

& \(\frac{5 }{k}\) = sec θ

\(\frac{1 }{k}\) = \(\frac{sec θ }{5}\)

Now,

10 ( k² - \(\frac{1 }{k²}\) )

= 10 ( (\(\frac{tan θ }{5}\))² - (\(\frac{sec θ }{5}\))² ) 

= 10 ( (\(\frac{tan² θ }{25}\)) - (\(\frac{sec² θ }{25}\)) )

{ we know, sec² θ  - tan² θ  = 1 }

= -\(\frac{2 }{5}\)