Factoring Trigonometric Expressions...2

[nycmathguy]

Section 5.1

Question 24

sec^4 x - tan^4 x

(sec^2 x - tan^2 x)(sec^2 x + tan^2 x)

(sec x - tan x)(sec x + tan x)(sec^2 x + tan^2 x)

Can this be further simplified?

 

[MathLover1]

sec^4 (x) - tan^4 (x)

=(sec^2 (x))^2 - (tan^2 (x))^2

=(sec^2 (x) - tan^2 (x))(sec^2 (x) +tan^2 (x)).........use identity tan^2 (x)=sec^2 (x) -1

=(sec^2 (x) - (sec^2 (x) -1))(sec^2 (x) +sec^2 (x) -1)

=(sec^2 (x) - sec^2 (x) +1)(2sec^2 (x) -1)

=1*(2sec^2 (x) -1)

=2 sec^2(x) - 1

 

[nycmathguy]

sec^4 (x) - tan^4 (x)

=(sec^2 (x))^2 - (tan^2 (x))^2

=(sec^2 (x) - tan^2 (x))(sec^2 (x) +tan^2 (x)).........use identity tan^2 (x)=sec^2 (x) -1

=(sec^2 (x) - (sec^2 (x) -1))(sec^2 (x) +sec^2 (x) -1)

=(sec^2 (x) - sec^2 (x) +1)(2sec^2 (x) -1)

=1*(2sec^2 (x) -1)

=2 sec^2(x) - 1

I forgot about the identity for tan^2 x. This is why I got stuck. However, is my work correct right up to
sec x - tan x)(sec x + tan x)(sec^2 x + tan^2 x)?
I will try a few more on my meal break tonight. Of course, we will continue with Section 5.1 tomorrow into the weekend. Like I said before, math is just a hobby for me at 56. No test to study for. No teacher to answer to.