If 5x = sec(θ) and 5/x = tan(θ), then x = sec(θ)/5 and 1/x = tan(θ)/5.
Then
5 (x² - 1/x²) = 5 ((sec(θ)/5)² - (tan(θ)/5)²)
… = 5 (sec²(θ)/25 - tan²(θ)/25)
… = sec²(θ)/5 - tan²(θ)/5
Recall that
sec²(θ) = 1 + tan²(θ)
Then
5 (x² - 1/x²) = (sec²(θ) - tan²(θ))/5 = 1/5
