take @=theta

sec@+tan@=x

sec@.tan@=y

eliminate @ with proper solution

sec@tan@ = y ...........1

sec@+tan@ = x         ...............2

squaring eq 2

(sec@)² + (tan@)² + 2sec@tan@ = x² 

1+2(tan@)²+2sec@tan@ = x²                     (sec²@ = 1+tan²@)

1+ 2(tan@)² + 2y = x²                 ....................3            (from eq 1)

squaring eq 1

(sec@)²(tan@)² = y²

 (1+tan²@)(tan²@) = y²            ...................4

substituting tan²@ from eq3 in eq4

( x²-2y +1)(x²-2y-1) = 4y²

 (x²-2y)² - 1 = 4y²

 x⁴  -  4x²y  = 1

this is the required relation