Final answer:
To simplify the expression: 6x(secx tanx) -sec⁴x(secx tanx), factor out the common term (secx tanx) and simplify using the fact that sec²x = 1 + tan²x.
Step-by-step explanation:
To simplify the expression: 6x(secx tanx) -sec⁴x(secx tanx), we can factor out the common term (secx tanx). This leaves us with (secx tanx)(6x - sec³x). Now, we can rewrite sec³x as (sec²x)(secx). Substituting this back into the expression, we have (secx tanx)(6x - (sec²x)(secx)).
Expanding further, we get (secx tanx)(6x - sec³x) = (secx tanx)(6x - sec²x)(secx). Now, we can simplify this expression by using the fact that sec²x = 1 + tan²x. Substituting this in, we get (secx tanx)(6x - (1 + tan²x))(secx).
Simplifying even further, we obtain (secx tanx)(6x - 1 - tan²x)(secx) = sec³x tanx(6x - 1 - tan²x). And this is the final simplified expression.