A community for students.
Here's the question you clicked on:
 0 viewing
VeroZarate
 3 years ago
Verify that cot x sec4x = cot x + 2 tan x + tan3x
VeroZarate
 3 years ago
Verify that cot x sec4x = cot x + 2 tan x + tan3x

This Question is Closed

VeroZarate
 3 years ago
Best ResponseYou've already chosen the best response.0so i know cotx is cos/sin

VeroZarate
 3 years ago
Best ResponseYou've already chosen the best response.0idk if we can change the tan though to sin/cos?

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0this one is hard but try this

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0first divide everything by cot x

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0\[\sec^{4} x = 1 + 2\frac{ \tan x }{ \cot x } + \frac{ \tan^{3} x }{ \cot x }\]

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0and we know that \[\frac{ \tan x }{ \cot x } = \frac{ \frac{ \sin x }{ \cos x } }{ \frac{ \cos x }{ \sin x } } = \frac{\sin^{2} x }{ \cos^{2} x } = \tan^{2} x\]

VeroZarate
 3 years ago
Best ResponseYou've already chosen the best response.0would it apply to the second one but it would be (sin^3x/cos^3x)/cosx/sinx ?

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0that would simplify it to: \[\sec^{4} x = 1 + 2 \tan^{2} x + \tan^{4} x\]

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0yeah the second one is: \[\frac{ \tan^{3} x }{ \cot x } = \frac{ \tan x }{ \cot x } * \tan^{2} x = \tan^{2} x * \tan^{2} x = \tan^{4} x\]

VeroZarate
 3 years ago
Best ResponseYou've already chosen the best response.0where did you get the second tan^2x

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0I divided tan^3 x into tan^x * tan^2 x

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0dw:1357511455432:dw

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0So this is why we have the following equation so far: \[\sec^{4} x = 1 + 2 \tan^{2} x + \tan^{4} x\]

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0next step is to subtract both sides by tan^4 x

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0\[\sec^{4} x  \tan^{4} x = 1 + 2 \tan^{2} x\]

VeroZarate
 3 years ago
Best ResponseYou've already chosen the best response.0and divide both side by2?

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0no there is a trick here now. notice the left side is the difference of two squares

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0\[\sec^{4} x  \tan^{4} x = (\sec^{2} x)^{2}  (\tan^{2} x)^{2}\]

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0remember that a^2  b^2 = (a  b)(a + b) so we can now write the left side as: \[\sec^{4} x  \tan^{4} x = (\sec^{2} x  \tan^{2} x)(\sec^{2} x + \tan^{2} x)\]

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0we kind of did the reverse of FOIL, we factored it

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0So now we have: \[(\sec^{2} x  \tan^{2} x)(\sec^{2} x + \tan^{2} x) = 1 + 2 \tan^{2} x\]

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0but do you remember the identity sec^2 x = tan^2 x + 1

VeroZarate
 3 years ago
Best ResponseYou've already chosen the best response.0yes but there is the 2?

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0we'll get to that :) so if sec^2 x = tan^2 x + 1 the if we subtract tan^2 from both sides we get sec^2 x  tan^2 x = 1

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0\[(\sec^{2} x  \tan^{2} x)(\sec^{2} x + \tan^{2} x) = 1 + 2 \tan^{2} x\] becomes \[(1)(\sec^{2} x + \tan^{2} x) = 1 + 2 \tan^2{x}\]

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0No we factored \[\sec^{4} x  \tan^{4} x \] into \[(\sec^{2} x  \tan^{2} x)(\sec^{2} x + \tan^{2} x)\] because it is a difference of sqaures

VeroZarate
 3 years ago
Best ResponseYou've already chosen the best response.0ok so when you subtract tan^2x?

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0There is a trig identity: \[\sec^{2} x = 1 + \tan^{2} x\] i am saying, if we rewrite it as: \[\sec^{2} x  \tan^{2} x = 1\] then the left factor on our left hand side reduces to just 1: \[(\sec^{2} x  \tan^{2} x)(\sec^{2} x + \tan^{2} x) = (1)(\sec^{2} x + \tan^{2} x) \]

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0so we are now left with \[\sec^2{x} + \tan^{2} x = 1 + 2 \tan^{2} x\]

VeroZarate
 3 years ago
Best ResponseYou've already chosen the best response.0ok you had lost me. now i get it sorry

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0its a complicated problem compared to the last one o_o

VeroZarate
 3 years ago
Best ResponseYou've already chosen the best response.0yes it is. i still have to do 5 more assignments. lol

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0so now we have \[\sec^2{x} + \tan^{2} x = 1 + 2 \tan^{2} x\] can you see the next step ?

VeroZarate
 3 years ago
Best ResponseYou've already chosen the best response.0substitute the sec^2x with 1+ tan^2x?

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0we could, or we could save a step and just subtract tan^2 x from both sides

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0if we subtract tan^2 x from both sides we get \[\sec^{2} x = 1 + \tan^{2} x\]

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0which is an identity

VeroZarate
 3 years ago
Best ResponseYou've already chosen the best response.0but how does this equal to cotxsec^4x?

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0the whole idea is to reduce the original equation into a trig identity

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0if you follow the steps then that's precisely what we did

VeroZarate
 3 years ago
Best ResponseYou've already chosen the best response.0to verify that it equals on both sides?

binarymimic
 3 years ago
Best ResponseYou've already chosen the best response.0if you manipulate an equation using algebra then whatever you end up with is equal to what you started with.

VeroZarate
 3 years ago
Best ResponseYou've already chosen the best response.0oh ok thanks so much!
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.