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Here are the differences between the symbols.

tan x |
≡ | sin x tan x | again, this is true for all values of x |

≡ means "is identical to" or "is equivalent to". This shows that one side of the symbol is an alternative form to the other. You notice that x here is on both sides. The ≡ may mean that you can use one equation or the other. You could also think of it as showing another way of writing an equation.

sin

≜ means "is defined to be" so it shows what one thing means

The := simply means that a specific value is assigned to the variable.

This just means that it is true for certain values. In this case, n = ± 8

This just expresses a mathematical fact

In conclusion, you could use = for all the above if you wish.

Also some people use any one of the following symbols for "is defined by" =: := ≡ :⇔ ≜ ≝ ≐

It is not uncommon to use one symbol to mean different things for example, ≡ can be used in modulo arithmetic, 5 ≡ 2 mod 3 (which is congruence) and I have seen := used for "is defined by".

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