Trig identities. Is it wrong to square both sides of the identity when you try to prove LS = RS?

Question

If yes then why?

Answer 1

The problem with squaring both sides is that you can potentially add values for solutions. Take this example:

x = 1

Let's square both sides, to get

x^2 = 1

Then, let's solve this. x^2 = 1 means x^2 - 1 = 0, which means (x-1)(x+1) = 0, so x = 1 or x = -1. Squaring the LHS and RHS added a value that wasn't meant to be added!

The same goes for trigonometric identities.

Answer 2

It can be wrong, yes.
When you are trying to prove A=B, many people make the mistake of writing something like:
A=B
therefore A+5 = B+5
therefore..
therefore 1=1
which is true, therefore we have proven A = B.

Thats very bad mathematics. It looks like you have assumed that A = B and proven that 1 = 1 - whats the point of that? It is much better to start with 1=1, and work backwards to prove that A = B.

You can get away with writing it like the above, as long as you replace 'therefore' by 'if, and only if'. For example, if A=B then A+5 = B+5, and vice versa.
That doesn't always work with squaring though:
-1 = 1
(-1)^2 = (1)^2
1 = 1
That isn't a proof that -1 = 1.

If you know that both sides are always positive, or both sides are negative, you'll be fine - otherwise, it may not be a good idea.

(edit - the first two posters don't seem to know much about proofs :) You're trying to prove something - yet they say you are assuming its true, and proving it? That doesn't make sense.)

As a specific example, I can "prove" that cos x = sqrt(1 - (sin x)^2).
Square both sides:
(cos x)^2 = 1 - (sin x)^2
(cos x)^2 + (sin x)^2 = 1
true.

However, the original statement was false (it sometimes is that, sometimes is -that).

Answer 3

You can square both sides safely.

If you take the square root of both sides, don't forget the positive and negative roots. Puggy shows the issue. If an expression is true to begin with, its square is true. You can square both sides, but then don't go taking the square root carelessly.

Of course, you are assuming the identity true no matter what you do to it. When you "prove" an identity, you assume it is true, manipulate it in legal algebraic ways, and try to arrive at an obvious identity such as

cos(x) = cos(x)

Answer 4

Philosophically, yes, because you are assuming the identity is already true if you square both sides.

Answer 5

allow's seem on the first one: the definition of sec x is: sec x = a million/ cos x , hence: sinx secx = sinx * a million/ cosx = sinx/cosc and it truly is the definition of tanx, i.e sinx/cosx = tanx further use the definition of secy = a million/snug, and coty=snug/siny: ==> siny secy coty = siny a million/snug snug/siny = siny (snug/snug) a million/siny = siny/siny = a million I sparkling up the terrific one and also you may want to do the different 2 further: with the aid of definition: cotx = a million/tanx, so tanx(a million+cotx) = tanx (a million+a million/tanx) , multiply tanx contained in the words interior parantesis you get: tanx (a million+a million/tanx) = tanx + a million in case you imagine the answer is functional you may settle for it, i choose the point i'm getting out of your popularity, thanks lots.