Which Of The Following Equations Are Identities

Which of the Following Equations Are Identities? A Plain-English Guide

You ever stare at a list of equations and just know* one of them is true no matter what you plug in — but the rest? Total coin flips. That's the whole identity problem in a nutshell. Some equations hold for every value of the variable. Others only work for one or two. Telling them apart is less about memorizing rules and more about training your eye to spot the difference.

Let's walk through what makes an equation an identity, what makes it not one, and how to actually check without second-guessing yourself.

What Is an Identity, Really?

An identity is an equation that's true for every* value of the variable — not just some, not just one. Every. Single. Value.

The classic example is something like sin²(x) + cos²(x) = 1. Which means you can plug in any angle, any number, any weird value, and it holds. Always. That's why people call it a Pythagorean identity — it's not a thing you solve, it's a thing you trust*.

Contrast that with a regular equation. On the flip side, sin(x) = 0. In practice, 5 has solutions, sure — but it's only true at specific values (like π/6, 5π/6, and so on). It's a conditional equation, not an identity. On top of that, the difference isn't about how complicated it looks. It's about scope: is the equation always true, or just sometimes?

Here's the thing — most textbooks throw around the word "identity" like it's obvious, but in practice, students often confuse three different things: identities, contradictions (equations that are never true), and conditional equations (true for some values). Once you separate those, the problem gets way easier.

Why the Confusion Happens

Part of the problem is that some identities look* like they could be conditional. Because of that, at first glance you might think, "Okay, is that true for x = π/2? So technically the equation doesn't hold at x = π/2. Practically speaking, take tan(x) · cos(x) = sin(x). In practice, " And yeah — that one's a trap, because tan is undefined there. Does that disqualify it as an identity?

This is the part most students get hung up on. Strictly speaking, an identity holds for all values in the domain of both sides. If a value makes one side undefined, it's usually just excluded from the consideration. Still, in practice, though, that's a fine point that varies by instructor. What matters more is whether the equation reduces to something obviously true across the board — and tan(x)cos(x) does, because tan(x) = sin(x)/cos(x), so multiplying by cos(x) just gives you sin(x).

Knowing that one trick saves a lot of time.

Why It Matters (Beyond the Test)

Honestly, most people ask "which of the following equations are identities" because they're in a trig class, probably studying for a midterm. Fair enough. But identities show up in places that don't feel like math class at all.

Physics uses them constantly. Consider this: calculus leans on them too — when you integrate and need to simplify before you can move on, you're basically using trig identities as a toolkit. And any time you derive an equation that has to hold regardless of initial conditions, you're leaning on identities. Engineering, computer graphics, signal processing — they all build on the same handful of relationships.

So when you learn to spot an identity quickly, you're not just acing a quiz. You're training a skill you'll lean on whenever you're working with formulas and need to know whether a step is always* valid or only sometimes* valid. That distinction is everything in technical work.

How to Tell If an Equation Is an Identity

There's no single trick, but there's a reliable process. Here's how I'd approach it.

Step 1: Simplify One Side Until It Matches the Other

Take the equation and pick whichever side looks messier. Rewrite everything in terms of sine and cosine if you're dealing with trig. Also, cancel factors. Combine fractions. Your goal is to make both sides look identical.

To give you an idea, say you see: (1 − cos²(x)) / sin(x) = sin(x). So the left becomes sin²(x) / sin(x) = sin(x). But 1 − cos²(x) is just sin²(x) — that's a Pythagorean identity right there. Identical sides. The left side looks busier, so start there. But cancel the sin(x) (assuming sin(x) ≠ 0 for the moment) and you're left with sin(x) = sin(x). Identity confirmed.

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This is the workhorse method. It works for maybe 70% of the equations you'll see on a quiz.

Step 2: Look for "Red Flag" Values

Some equations look like identities but blow up at certain points. Take cot(x) · sin(x) = cos(x). Also, simplify cot(x) to cos(x)/sin(x), multiply by sin(x), and you get cos(x). Think about it: looks great. But what about x = 0? That's why cot(0) is undefined, so the left side doesn't exist there while the right side equals 1. Does that make it "not an identity"?

Depends on the context. But most textbooks will say yes, it's an identity on its domain. A few strict instructors will mark it wrong. The smart move is to know the convention your class uses — but in real life, this kind of equation is treated as an identity because it holds wherever it's defined.

The bigger red flag is when an equation simplifies to something like 0 = 1, which means it's a contradiction — true for no values. Or 0 = 0, which is true for everything (identity). Anything in between means it's conditional.

Step 3: Test Specific Values When You're Stuck

If simplification isn't working, plug in easy values and see what happens. Also, x = 0, x = π/2, x = π. Run the numbers.

• If the equation holds for all three, it's almost certainly an identity.
• If it fails on one, it's probably conditional or has a domain issue.
• If it fails on all of them, it's likely a contradiction.

This isn't a proof, but it's a quick way to check your work or eliminate options on a multiple-choice test. Most "which of the following" questions are designed so that a couple of values will knock out the wrong answers fast.

Step 4: Watch for Algebraic Sleight of Hand

Some "identities" aren't really identities at all — they're true only because someone multiplied by zero or divided by something that could be zero. Think about it: if you find yourself canceling an expression, ask: could that expression be zero? If yes, you've narrowed the domain, and the equation is technically not an identity over all reals.

This is the trickiest part, because the algebra looks clean. You cancel, you simplify, you declare victory. But the equation only worked because the thing you canceled was non-zero. That's not an identity — it's a conditional statement with extra steps.

Common Mistakes People Make

The biggest one? Identity means equivalent always*, with the domain caveat. Worth adding: treating "it simplified" as the same thing as "it's an identity. Think about it: " Simplification shows that two sides are equivalent* under certain conditions. Confusing these two is how students lose points on otherwise solid work.

Another one: assuming that if an equation involves trig functions, it must be an identity. Nope. sin(x) = x is a conditional equation, and a famous one at that. The trig part is irrelevant — what matters is the truth across all values.

And here's a subtle one. People sometimes think that if both sides of an equation give the same numerical answer when you plug in a value, the equation is an identity. But that's backwards reasoning. Day to day, you need the equation to hold for every* value, not just one or two. A single counterexample kills the identity claim.

Practical Tips That Actually Help

Memorize the core identities — the Pythagorean ones, the sum and difference formulas, the double angle formulas. Not because you need them on a flashcard, but because the moment you see a complex trig expression, you should be able to spot which identity is hiding in there. The recognition is what saves time.

When given a list of equations and asked to pick the identities, start with the easy ones. Knock out the ones that obviously have solutions (sin(x) = 0, for example — that's not an identity, that's a problem to solve). Then tackle the rest using the simplification approach.

If you're allowed a calculator, use it to test values — but only as a sanity check, never as proof. The math has to hold up without the calculator too.

And finally, when in doubt, rewrite everything in sines and cosines.