Simplifying Trig Expressions Again

[burger]

I don't mean to ask too many questions but I only have three more problems I really need help with and thats it.

Instructions: Factor and simplify, assuming all denominators are nonzero.

1) Tan^4 X - Sec^4 X

I turned Tan into Sin^4 X / Cos^4 X and turned Sec into 1 / Cos^4 X then subtracted and got (Sin^4 X - 1) / Cos^4 X and I have no idea how to simplify it further. I'm guessing with a pythagorean identity but I have no idea how to apply it.

2) 4 Sec^2 X + 8 Sec X + 4

I factored this into (4 Sec X + 4)(Sec + 1) but yet again I'm not entirely sure how to simplify it further.

3) (6 Tan X Sin X - 3 Sin X) / (9 Sin^2 X + 3 Sin X)

This one is monstrous in my opinion. The only thing I can think of doing is takeing out the common factor in the denominator and turning tan into Sin / Cos and then distibuting in the numerator but after that I have no idea what to do.

 

[stapel]

1) tan⁴(x) - sec⁴(x)

Applying the difference-of-squares formula, we get:

. . . . .(tan²(x) - sec²(x))(tan²(x) + sec²(x))

Then use the fact that 1 + tan²(x) = sec²(x).

2) 4 sec²(x) + 8 sec(x) + 4

This is just a quadratic in secant:

. . . . .4 [sec²(x) + 2 sec(x) + 1]

. . . . .4 [sec(x) + 1]²

I don't think there's much else you can do with this.

3) [6 tan(x) sin(x) - 3 sin(x)] / [9 sin²(x) + 3 sin(x)]

Deal with the simple factoring first, naturally:

. . . . .[3 sin(x) (2 tan(x) - 1)] / [3 sin(x) (3 sin(x) + 1)]

. . . . .[2 tan(x) - 1] / [3 sin(x) + 1]

Then see if converting to sines and cosines helps at all. (It doesn't always, but it's usually a good thing to check.)

. . . . .[2 sin(x)/cos(x) - 1] / [3 sin(x) + 1]

. . . . .[(2 sin(x) - cos(x)) / cos(x)] / [3 sin(x) + 1]

. . . . .[2 sin(x) - cos(x)] / [cos(x) (3 sin(x) + 1)]

This doesn't seem to be leading anywhere promising. I think the last step before converting was probably as good as this will get.

Eliz.

 

[galactus]

I don't mean to ask too many questions but I only have three more problems I really need help with and thats it.

Instructions: Factor and simplify, assuming all denominators are nonzero.

1) \(\displaystyle Tan^{4}(x) - Sec^{4}(x)\)

Notice it's a difference of two squares?. \(\displaystyle a^{2}-b^{2}=(a+b)(a-b)\)

\(\displaystyle \L\\(tan^{2}(x)+sec^{2}(x))(tan^{2}(x)-sec^{2}(x))\)

Finish up?.

[quote:38nfa2n9]2)\(\displaystyle 4 Sec^{2}(x) + 8 Sec(x) + 4\)

This looks like quadratic, doesn't it?. Let \(\displaystyle u=sec(x)\) and factor.

3) \(\displaystyle \frac{(6 Tan X Sin X - 3 Sin X)} {(9 Sin^2 X + 3 Sin X)}\)

[/quote:38nfa2n9]

Start by factoring 3sin(x) out of the denominator and numerator:

\(\displaystyle \L\\\frac{3sin(x)(2tan(x)-1)}{3sin(x)(3sin(x)+1)}\)


EDIT: Sorry Stapel. Looks like I just duplicated what you posted.

 

[burger]

Thank you guys so much for helping me. I actually feel like I've learned something.