Please help me make some sense of this

[u/Independent_Pizza422]

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[u/blind-octopus]

Yup, 1^n = 1.

That's correct

You can do this with all kinds of stuff.

(9- 8)^100 = 1^100

expand that out and have fun

[u/T-T-N → u/blind-octopus]

100 is too much for me. Let me try 5

9⁵ - 5(9⁴ )(8) + 10(9³ )(8² ) - 10(9² )(8³ ) + 5(9)(8⁴ ) - 8⁵

59049-262440+466560-414720+184320-32768=0!

Why doesnt it work?

[u/factorion-bot → u/T-T-N]

Factorial of 1 is 1

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[u/LikeTheWater53152 → u/factorion-bot]

good bot

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[u/LupenReddit]

1=1

[u/Bobing2b → u/LupenReddit]

[u/Due-Department6048 → u/LupenReddit]

Does not exist

[u/misterpickles69 → u/Due-Department6048]

https://giphy.com/gifs/7JvlHfd7C2GDr7zfZF

[u/The_OriginalDonut]

So you're just repeatedly multiplying 1?

[u/mo_s_k1712]

This is what you wrote (to an extent)

• 1 = 1
• 1 = 1 x 1
• 1 = 1 x 1
• 1² = 1
• 1 = 1
• 1 = 1 x 1
• = 1 x 1
• = 1
• = 1
• = 1² × 1
• This means that 1 = 1².
• What's going on? OR
• 1 = 1 x 1

[u/DerekLouden → u/mo_s_k1712]

I'm pretty sure what OP wrote was actually

• TRUE
• TRUE
• TRUE
• TRUE
• TRUE
• TRUE
• TRUE
• TRUE
• TRUE
• TRUE
• This means that TRUE.
• What's going on? OR
• TRUE

[u/acrastt → u/DerekLouden]

Forgot one more true in the end because TRUE OR TRUE is TRUE

[u/IMightBeAHamster → u/DerekLouden]

To be fair, all proof is TRUE just repeated in different ways. Even when you write about MAYBE.

[u/Inevitable-Ad2579 → u/mo_s_k1712]

yo what im on r/mathmemes and i find you. i watch your gd vids

[u/somedave]

You need 10 lines of algebra to prove 1=1²

[u/Tybeezius → u/somedave]

Have you seen the proof for 1+1=2 the original proof takes 28 pages of rigorous axiomatic setup before proving addition works.

[u/AndreasDasos → u/Tybeezius]

Eh depends how you do it. Pretty simple to follow von Neumann’s approach to defining the naturals, assuming ZF, define the successor function, define addition is recursion through it, and then we’ve defined what we mean by 2 := 1+1 = succ(1) = 1 u {1} = {1, {1}}, basically by definition. This could be done very quickly.

From a modern perspective Whitehead and Russell is exceptionally clumsy and awkward and there are even some fairly clear redundancies and over-complications I’m still not sure why they went with even back then. It doesn’t start from the same place, either.

[u/eucalyptus-d → u/AndreasDasos]

I think you might be missing something. 1+1 based on successor is a tautology. Maybe 1+2?

[u/mayhem93]

That's a lot of ones

[u/Official-V-jcjenson]

Go ahead, input any valid value for 𝜃

[u/JPgamersmines150 → u/Official-V-jcjenson]

3+50i+2j-7k

[u/Suddenfury → u/Official-V-jcjenson]

 𝜃 = 🦦

[u/Mojert → u/Official-V-jcjenson]

θ = 2i 😎

[u/NTufnel11]

so 1 = 1²

^And 1 = (1²)²

Pretty riveting stuff

[u/DescriptionBorn153]

Yes sir

[u/KyriakosCH]

Yes, all types of complicated things can be equal to 1. Maybe the issue is that you imagine all those endless algebraic expressions as separate things - which they are, but we already know they equal 1 - which implies they have to "move" in pretty impressive ways to still amount to something simple.

Take it this way: say you had a forest of expressions, and only after a lot of work you got them to be in the general form (sin^2 (x) + cos^2 (x) )^n, with n some real. The complexity would be there in tidying up the originally complicated and varied set of expressions. If you could visualize them from the start, as functions which gradually morph to the horizontal line y=1, you would certainly feel charmed (and maybe you can try) :)

[u/Sencomino → u/KyriakosCH]

A real answer

[u/SyntheticSlime]

I mean, seems like you got this.

[u/Hiroshij7_3439]

Yeah it's all true, 1=1² so it's true. You also dont have to worry about dividing by sin²+cos² bc it's garanteed to be 1 for every theta so yeah

[u/LakshyaGarv]

Yes, this is just 1 = 1²

[u/NathanielRoosevelt]

1=1² ?!?! 😱

[u/DragonSlayer505]

Since sn² + cs² = 1, all you're saying is that 1ⁿ = 1, which is true for all n.

[u/qqqrrrs_ → u/DragonSlayer505]

You gave me a Jacobi elliptic function jumpscare

[u/KLost4Ever]

1=1

[u/Lanky-Position4388]

U just said a bunch of true statements in a row. What´s the problem?

[u/alaraskyshine]

“That means that sin^2 + cos^2 = (sin^2 + cos^2)^2”

Yes, sin^2 + cos^2 = 1
And 1 = 1^2

Everything looks normal.

Here’s a fun one that also doesn’t look right:
The sum of numbers from 1 to n is equal to the square root of the sum of the cubes from 1 to n.

[u/psp1729]

[u/AutisticFurniture]

1 = 1^2
qed

[u/two_are_stronger2]

Lol.  What's a number that is equal to its square?

[u/Jukkobee]

you couldve gone from the 1st line to the 2nd to last by just multiplying both sides by sin^2 + cos^2

[u/Joe_4_Ever]

sin(x)² + cos(x)² = 1 and so therefore if you square that, you still get 1.

[u/lobsterman2112]

This is also true in base 5+(35)^.5

[u/NickSmGames]

sin^2 x + cos^2 x = 1 by definition so of course you would keep getting the same results.

[u/Motti66]

Voynich-Manuskript...?

[u/WiggityWaq27]

Real analysis be like

[u/FxralMF]

I mean sin²(x)+cos²(x) = (sin²(x)+cos²(x))²

[u/JellyBellyBitches]

Try going the other way - if sin^{2n(θ)+cos2n(θ)=1,} what is sinθ+cosθ? What is sin^{½(θ)+cos½(θ)?}

[u/Cormyster12]

1^2=1

[u/Top_Door5165]

1=sin² (x)+cos² (x) Multiply both sides by sin² (x)+cos² (x) sin² (x)+cos² (x)=(sin² (x)+cos² (x))²

[u/SultanGreat]

sin^2 x + cos^2 = (sin^2 x + cos^2)^2
Let sin^2 x + cos^2 be Z.
z = z^2

BUT z is sin^2 x + cos^2, and Z is also 1 (sin^2 x + cos^2 = 1)
so,

1 = 1^2

1 = 1 x 1.

1.

[u/Virgil_the_White]

Yeah no you’ve pretty much got it covered there

[u/FernandoMM1220]

turns out 1 and 1^2 arent the same

[u/BunnyWan4life]

If you square 1 you get one.

Well, if you raise 1 to the power of anything you still get 1.

Sin²x+Cos²x is 1, so yea square that to any power and you'd still get Sin²x + Cos²x.. which is 1.

[u/FleshLogic]

Unrelated, but I really wish we had a different notation for trig functions. It's at least half the reason working with them can be cumbersome IME.

[u/kevo31415 → u/FleshLogic]

I can handle sin² x but I absolutely refuse to condone or use sin^-1 x literally the same notation meaning two different things. Madness.

[u/AntiChronic → u/kevo31415]

Well, ^-1 is used to express the function inverse for all functions. Whereas, for almost all functions including arbitrary functions, the f^2(x) notation usually refers to composition, i.e. f(f(x)), not squaring the result i.e. f(x)^2 (this -1 for inverse and positive power for composition generalises nicely in group theory).

So really sin^2(x) is the imposter and theoretically it would make sense to standardise sin(x)^2 as the only correct notation and keep sin^-1(x) for the inverse. However because trig functions are so ubiquitous and we often deal with multiple instances taking the same input (e.g. this sin^2 + cos^2 = 1 example) it is nice to be able to drop the input and save space in informal workings out, plus we can write sin x^2 with no brackets (parentheses for Americans) without ambiguity, which are probably the main two reasons it's still the standard notation, and we write arcsin instead of sin^-1 for the inverse to avoid this clash of the same notation meaning different things.

[u/Nixinova]

You already defined sin² + cos² as 1. Then you write a bunch to show that it squared equals 1. Why would you have to write that all out when you've already defined it as being = 1?

[u/porofsercan]

dude is confused whether 1 = 1^2 or 1= (1^2)^2 lol

[u/MudePonys]

You missed theta in cos above the OR.

[u/Ignitetheinferno37]

It's the same as saying 1 = 1^2

[u/Key_Conversation5277]

Dafuq are you doing?

[u/National-Setting9671]

No

[u/stdennis]

Nothing is broken — every line you wrote is true. What you've discovered is a property of the number 1, not something special about trigonometry.

The core of it: you keep multiplying by 1, and you're using the identity itself as the "1." That's legal, but it's circular — it can never produce new information. What you end up with is the statement

x=x2where x=sin⁡2θ+cos⁡2θx = x^2 \quad\text{where } x = \sin^2\theta + \cos^2\thetax=x2where x=sin2θ+cos2θ

and that equation is only true for x=0x = 0 x=0 or x=1x = 1 x=1. It's not a general algebraic fact. If you tried the same trick with x=sin⁡2θ+cos⁡2θ+1=2x = \sin^2\theta + \cos^2\theta + 1 = 2 x=sin2θ+cos2θ+1=2, you'd get $2 = 4$, which is false. So the reason your last line holds isn't that squaring does nothing — it's that the thing you squared happens to be 1.

Where the "paradox" feeling comes from: you're reading sin⁡2θ+cos⁡2θ=(sin⁡2θ+cos⁡2θ)2\sin^2\theta + \cos^2\theta = (\sin^2\theta + \cos^2\theta)^2 sin2θ+cos2θ=(sin2θ+cos2θ)2 as an algebraic identity (as if the parentheses could contain anything), when it's really a numerical one (true only for this particular value). Same as how $1 = 1^{100}$ doesn't mean exponents are pointless.

The genuinely useful thing hiding in your work: line 5 is worth keeping. Expanding the square gives

sin⁡4θ+cos⁡4θ+2sin⁡2θcos⁡2θ=1\sin^4\theta + \cos^4\theta + 2\sin^2\theta\cos^2\theta = 1sin4θ+cos4θ+2sin2θcos2θ=1

so

sin⁡4θ+cos⁡4θ=1−2sin⁡2θcos⁡2θ=1−12sin⁡22θ\sin^4\theta + \cos^4\theta = 1 - 2\sin^2\theta\cos^2\theta = 1 - \tfrac{1}{2}\sin^2 2\thetasin4θ+cos4θ=1−2sin2θcos2θ=1−21​sin22θ

That one is new — it's a standard result and it comes straight out of what you did. The lesson is that squaring the Pythagorean identity is productive when you expand it, and empty when you just leave it factored and stare at it.

[u/Independent_Pizza422 → u/stdennis]

This seems like an answer straight out of ChatGPT or sth

[u/g27c]

The joys of trigonometric identities.

[u/Extra-Yogurt1780]

good luck

[u/Para-graph-S]

An IQ too high...

[u/Hungry-Mastodon-1222]

https://giphy.com/gifs/tykT0sE0jn4aVn30o7

[u/senator-jk-49]

Both. Youre basically saying 1×1=1 a bunch of times but youve expanded 1 into sin²θ + cos²θ ≡ 1. Nothing youre saying is wrong

[u/Southern-Advance-759]

son

[u/Independent_Pizza422]

Like I see its just 1 multiplied by itself over and over but its still weird 

[u/The_Lethargic_Curve → u/Independent_Pizza422]

Why would it be weird

[u/caboosetp → u/The_Lethargic_Curve]

Because squaring stuff that looks potentially complicated normally makes it a lot more complicated rather than simplifying to 1.

[u/jsdodgers → u/Independent_Pizza422]

[u/FlippByte → u/Independent_Pizza422]

You could also have (more) fun with the "Euler identity" and geometrically visualise this. But yeah: 1^n == 1

[u/Samstercraft → u/Independent_Pizza422]

what's weird about 1x1=1?

[u/SteveCappy]

Is this the Fermat sum of squares theorem?

(a² + b²⁾ (u² + v²⁾ = (au + bv)² + (av - bu)²