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

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

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

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

binarymimic
 one year 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
 one year 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
 one year 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
 one year 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
 one year 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
 one year ago
Best ResponseYou've already chosen the best response.0where did you get the second tan^2x

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

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

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

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

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

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

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

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

binarymimic
 one year 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
 one year 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
 one year 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
 one year ago
Best ResponseYou've already chosen the best response.0ok so when you subtract tan^2x?

binarymimic
 one year 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
 one year 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
 one year ago
Best ResponseYou've already chosen the best response.0ok you had lost me. now i get it sorry

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

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

binarymimic
 one year 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
 one year ago
Best ResponseYou've already chosen the best response.0substitute the sec^2x with 1+ tan^2x?

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

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

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

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

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

binarymimic
 one year 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
 one year ago
Best ResponseYou've already chosen the best response.0oh ok thanks so much!
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