Trig Identities

[granite stone]

For example:

$\frac{\sec t}{\cos t}-\frac{\tan t}{\cot t}=1$

[leaden thunder]

bondalton

[granite stone]

is equivalent to the following

[
\frac{\sec t\cot t-\cos t\tan t}{\cos t\cot t}=1
]

[leaden thunder]

bondalton

[granite stone]

Which is equivalent to

[\sec t\cot t-\cos t\tan t=\cos t\cot t]

[leaden thunder]

bondalton

[granite stone]

Now, we know that $\sec t=1/\cos t$ and $\tan t=\sin t/\cos t$, so that we can rewrite the latter as follows

[leaden thunder]

bondalton

[granite stone]

[\frac{1}{\cos t}\frac{\cos t}{\sin t}-\cos t\frac{\sin t}{\cos t}=\cos t\frac{\cos t}{\sin t}]

[slow owl]

always write trig functions in terms of sine and cosine when trying to derive/prove identities

the main identity you will need other than that is sin^2 + cos^2 = 1

at least for more basic identities

[leaden thunder]

bondalton

[ivory blade]

sorry but everything above the Pythagorean identity just made it more complicated

idk how to explain it

like okay so like

like i have all the identites

my math teacher wants us to use them to figure out how to prove these identities

but everything that bondalton just did made it all more confusing

[slow owl]

all I mean is that whenever you see tan, cot, sec, or csc you should rewrite them like this

[ivory blade]

thats what im doing

[slow owl]

I'll walk you through one of the exercises using that method if that's cool with you

[ivory blade]

thats how were told to do it

reciprical identities

[slow owl]

we'll do this one step by step

and you can ask any questions that are points of confusion

[ivory blade]

okay so Sec is 1/cos

[slow owl]

yes

[ivory blade]

and cos is just.. cos/1

okay but wait im dividing fractions so dont i swap it

[slow owl]

this is what you have so far

how would you evaluate that

[ivory blade]

okay thats what i thought

[slow owl]

it's correct just so you know

[ivory blade]

okay so then

Sin/Cos

/cos/sin

[slow owl]

for tangent ✅

yes

do you know how to simplify them further or would you like some review on fraction division

[ivory blade]

idk my teacher says for fractions to take what your dividing, switch the values then multiply them

but i dont know because it looks different than the other stuff

like i over think everything because of how it looks even though it probably has the same rules of algebra and crap

[slow owl]

the way they're usually done is "keep change flip"

so for the numerator (top of the fraction) you "keep" it the same
for the division symbol you "change" it to a multiplication symbol
and for the denominator (bottom of the fraction) you "flip" it into its reciprocal form

[ivory blade]

so pretty much what i just said :/

[slow owl]

yeah

[ivory blade]

so 1/cos x 1/cos

which is just 1/cos

or did i just butcher it

[slow owl]

IMO this is the better way to approach fraction division, since division is a multiplicative process

its 1/cos^2

[ivory blade]

why

[slow owl]

when you do fraction multiplication you multiply across

[ivory blade]

yeah

i know

[ivory blade]

okay well i thought it stayed the same because its on the bottom.

[slow owl]

nah it gets squared

[ivory blade]

okay

my bad

im stupid

[slow owl]

nah you're all good, it's part of the learning process.

[ivory blade]

i should know how to do this im in grade 11 math and Ive already done this with TONS of other shit

but just because it looks different i screw it all up

[bright canyon]

Which number are you on now?

[slow owl]

for the next term we get it into this form

[ivory blade]

okay im not done wait/

let me do it

okay. im here/

is it 1/cos2 - sin2/cos2

[slow owl]

yes

[ivory blade]

okay

idk what to do

[bright canyon]

Combine it to one fraction

[ivory blade]

i dont know how its different than before

[slow owl]

The denominators are the same so you can rewrite it like this

(1 - sin2)/cos2

[ivory blade]

oh right

see there we go simple algebra that i screwed up because my brain cant work when its with diffierent like things

wow an because 1 - sin2 is there its cos2 meaning its 1

[slow owl]

Correct

[bright canyon]

There’s another way you can prove it also

[slow owl]

Working with math that makes you feel like that is where the most improvement happens from my experience.

[bright canyon]

Divide the basic identify by cos^2(x)

[ivory blade]

i think ill stick to what i did before

[bright canyon]

Ik Im just saying

[ivory blade]

okay so 6

[ivory blade]

(1/cos2 + sin2/cos2)(1/cos2 - sin2/cos2)

[bright canyon]

So what can you multiply the entire thing by

[slow owl]

NICE

I was actually about to bring up difference of 2 squares

[ivory blade]

what

[bright canyon]

To simplify it

[ivory blade]

okay so now i do the thing

where i like

[slow owl]

(1/cos2 - sin2/cos2) is actually the same term we had in the number 4

[ivory blade]

oh

wow

wait

oh now i just add fractions

(sin2/cos2)(1 - sin2/cos2)

[slow owl]

close, the first term should be((1 + sin2)/cos2)

[ivory blade]

wait

\

wait

wait

i thought i was multiplying them so i got rid of the 1

omg

[slow owl]

all good

[ivory blade]

okay now lets look for identities

1 - sin2

cos2

[slow owl]

yes

so (1 - sin2/cos2) is just cos2/cos2

[ivory blade]

yes

which is 1

[slow owl]

yes

[ivory blade]

what about the first fractionw we have

[slow owl]

rewrite the right hand side in the same way you did to the left hand side

[ivory blade]

what

what do you mean

[slow owl]

rewrite this part

or alternatively try converting this back into tan and sec

[ivory blade]

sin2/cos2 + 1/cos2

[slow owl]

yes

[ivory blade]

uh

why did i do thus

[slow owl]

because now we have
((1 + sin2)/cos2)(1) = sin2/cos2 + 1/cos2

they're the same thing already

[ivory blade]

how

1 isnt equal to 1/cos2

[bright canyon]

After this question, you should see if he can figure it out on his own

[slow owl]

combine sin2/cos2 + 1/cos2 into one fraction

[ivory blade]

wait

[slow owl]

((1 + sin2)/cos2)(1) <-- this (1) is just to show that we found that one term was equal to 1.

[ivory blade]

1 + sin2y/cos2y

[slow owl]

yes

[ivory blade]

but

nbm

omg

okay ima do 7

[slow owl]

this is what we have right now

[slow owl]

nice

[ivory blade]

i wll let you know

[slow owl]

I'm gonna go finish cooking my food, I'll come back and see what you did

[ivory blade]

oh shit theres a 2 now

its different

i dont know what to do with the 2

[bright canyon]

Wdym

[ivory blade]

the 2

beside the tangent

[bright canyon]

Factor the left first

[slow owl]

^ start with the left side and it should become apparent

[ivory blade]

yeah how do i know the side to start with

(1/1 - Tan/1)(1/1 + Tan/1)

[bright canyon]

It depends

[ivory blade]

On??

[slow owl]

I think these exercises are specifically designed so that you should focus on rewriting the left side

[ivory blade]

no no no

our teacher said

to do the side that looks the hardest

thats it

[bright canyon]

Lol

[ivory blade]

i dont find it funny

[bright canyon]

You should be able to do half of these in your head

If not more

[ivory blade]

yeah well sorry im not as great at math as you are

[bright canyon]

This is a mix of critical thinking and memorization

[slow owl]

mmmm, yeah. I mean in theory you can always do it from either side. but usually one side is much easier to engineer into the other. so "more complicated" is sort of a general rule, though it becomes subjective like with number 7. because I agree that both sides are complicated.

[ivory blade]

okay so was my thing before right

[slow owl]

you definitely CAN rewrite both sides though

[bright canyon]

If you want to do right first rewrite sec^2 theta

[slow owl]

I'd definitely start with foiling out (1 - tan)^2 first

[ivory blade]

YEAH WAS my thing correct???

[slow owl]

no

[ivory blade]

why

\

[slow owl]

difference of squares is for when you have something like this

[bright canyon]

Where does this even come from because thats definitely not what the square says

[slow owl]

this is what we're dealing with

[bright canyon]

The speed that you send those is kinda insane

[slow owl]

lol 2 monitors

[ivory blade]

I dont fucking know

[bright canyon]

Think of it like this what do you get when you square the equation a+b=x

[ivory blade]

i dont know

[slow owl]

what does this expand out to?

using FOIL if you're familiar

[ivory blade]

yeah i know what FOIL is

dude im over thinking everything you guys put at me

this isnt helping

[slow owl]

I just replaced tan with x, since they work the same algebraically and I thought maybe using the familiarity of x might help.

[bright canyon]

Now I understand my calc teacher’s disappointment looking at the class

[ivory blade]

wow thanks

really appreciate it

[bright canyon]

You’re welcome

foil the problem what do you get

[slow owl]

using this as a template

[ivory blade]

Thats what i FUCKING HASD

HAD

[slow owl]

try this

[bright canyon]

Lmao

[ivory blade]

dude

I had that before

oh i had a + instead

[slow owl]

yeah

[ivory blade]

look i dont want to do this with Loukes in here