# mathematics equations concept

##### Full Member
Since it is always true that sin^2(x) + cos^2(x) = 1, you can always replace it with 1. Always! You need no additional information to make that permissible.

Why would you even imagine you couldn't??
I have no idea why I imagine that I couldn't , but I frankly face that problem, I know there's identity sin^2(x) + cos^2(x) = 1 , but ! while solving like if I have another problem like
Sin(x)^2 + Cos(x)^2 = X^2+Y^2+Z^2 .. so here i STUCK although I know that there's
sin^2(x) + cos^2(x) = 1 but the question why I could assign that?! here what's exactly I'm facing while solving question ...

maybe because sin^2(x) + cos^2(x) = 1 it's not given explicitly like sin^2(x) + cos^2(x) = 1 ..... so I get mislead while it's given with other equation like Sin(x)^2 + Cos(x)^2 = X^2+Y^2+Z^2....how can I solve that problem of thinking ?!!

#### JeffM

##### Elite Member
I have no idea why I imagine that I couldn't , but I frankly face that problem, I know there's identity sin^2(x) + cos^2(x) = 1 , but ! while solving like if I have another problem like
Sin(x)^2 + Cos(x)^2 = X^2+Y^2+Z^2 .. so here i STUCK although I know that there's
sin^2(x) + cos^2(x) = 1 but the question why I could assign that?! here what's exactly I'm facing while solving question ...

maybe because sin^2(x) + cos^2(x) = 1 it's not given explicitly like sin^2(x) + cos^2(x) = 1 ..... so I get mislead while it's given with other equation like Sin(x)^2 + Cos(x)^2 = X^2+Y^2+Z^2....how can I solve that problem of thinking ?!!
Again, you make up some problem to confuse yourself.

expression one = expression two.

All that means is that the two expressions represent the same numeric value.

$$\displaystyle cos^2(x) + sin^2(x) = \text {some expression.}$$

But no matter what x is $$\displaystyle cos^2(x) + sin^2(x) = 1.$$

So $$\displaystyle cos^2(x) + sin^2(x)$$ and 1 are just different names for the exact same numeric value.

And because you have stipulated that $$\displaystyle cos^2(x) + sin^2(x) = \text {some expression.}$$,

then some expression is just a third name for the exact same numeric value, namely 1.

The equal sign simply means "has the same numeric value."

You asked this exact same question using x, y, and z weeks ago.

Here is a BASIC rule of algebra: $$\displaystyle x = y \text { and } y = z \implies x = z.$$

Learn it and move on.

#### Ryan\$

##### Full Member
thanks alot but for instance lets assume I have like this
np*(5+y)
and np is equal to ni^2/C
then I know that's np=ni^2/C BUT once I solve an equation like this np*(5+y) it rings in my head that
np = ni^2/C , but I don't assign it because I ask myself who said that I'm allowed to "assign" .. yea they are equal but who said that I could assign in the equation np*(5+y) ..

I hope I explained my problem, you know it rings in my head that np=ni^2/C .. but I'm not assigning it although it rings in my head ni^2/C .. idk what's that approach? am I facing that problem alone? or maybe my iq isn't ... ? but it rings for me that np=ni^2/C .. but not assigning that in my equation ..

#### JeffM

##### Elite Member
thanks alot but for instance lets assume I have like this
np*(5+y)
and np is equal to ni^2/C
then I know that's np=ni^2/C BUT once I solve an equation like this np*(5+y) it rings in my head that
np = ni^2/C , but I don't assign it because I ask myself who said that I'm allowed to "assign" .. yea they are equal but who said that I could assign in the equation np*(5+y) ..

I hope I explained my problem, you know it rings in my head that np=ni^2/C .. but I'm not assigning it although it rings in my head ni^2/C .. idk what's that approach? am I facing that problem alone? or maybe my iq isn't ... ? but it rings for me that np=ni^2/C .. but not assigning that in my equation ..
Just stop making up these goofy examples. They frequently make no sense.

np * (5 + y) is an expression. It is NOT an equation. An algebraic expression has little meaning on its own.

"Ringing in your head" and "assigning" make no sense except perhaps in describing to yourself your own pschology. We are not psychiatrists. If two expressions have the same numeric value, it should be clear that it makes absolutely no numeric difference which you use. You can use whichever is more convenient for your purposes.

#### Harry_the_cat

##### Senior Member
Here's a hypothetical question for you Ryan. Do these sentences mean the same thing?
1. I find mathematics difficult.
2. I find mathematics hard.

Yes they do, because we know that the word "difficult" and the word "hard" are synonyms, ie have the same meaning in this context. So we can interchange them because we know they mean the same thing. The meaning of the sentence doesn't change. We could say difficult = hard.

It's the same deal in maths. If we know that sin^2(x) + cos^2(x) = 1, which it ALWAYS does, then we can interchange them. Wherever we see sin^2(x) + cos^2(x) in an equation or an expression, we can replace it with 1.

#### Jomo

##### Elite Member
You need to understand what equal means.
If klegjagkj= hjsaoaf then whenever you see klegjagkj you can replace it with hjsaoaf and whenever you see hjsaoaf you can replace it with klegjagkj.

Now in a given problem you are told for example that x+y = 9. So in this problem if you see x+y you can replace it with 9 and if you see 9 you can replace it with x+y.

Now x+y=9 is NOT an identity equation as it is NOT always true. However you know that 2+3=5. So if in your work you have 2+3 I bet you would replace it with 5. Now just like 2+3=5 we have sin^2(x) + cos^2(x) = 1 ALWAYS. So you can replace sin^2(x) + cos^2(x) =with 1 without any real thinking just as you would replace 2+3 with 5.

Just understand that just because you know very well that 2+3=5 and maybe don't really know why sin^2(x) + cos^2(x) = 1 they are still identities non-the-less.

So much is 7 + 4(sin^2(x) + cos^2(x))

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