What an Identity Proof Really Is (and What It Is Not)
An identity proof is not “solving for a variable.” It is a demonstration that two expressions are equivalent for every angle where both sides are defined. The most reliable workflow is to transform one side at a time until it matches the other side exactly, while keeping the other side unchanged as a target.
Think of the unchanged side as a “destination.” Every step you write should be a legal algebraic/trig rewrite that keeps the expression equivalent on its domain.
The One-Side Method (Core Workflow)
- Step 0: Choose a side to work on. Usually pick the more complicated side (more terms, mixed functions, nested fractions).
- Step 1: Rewrite using simple identities first. Convert quotients/reciprocals, clear complex fractions, factor, combine over a common denominator.
- Step 2: Use Pythagorean substitutions when you see
1 - sin^2,1 - cos^2,1 + tan^2,1 + cot^2. - Step 3: Use sum/difference or double-angle only when the structure suggests it. For example, paired terms like
sin x cos yandcos x sin y, or expressions likesin^2 x,cos^2 x,2sin x cos x. - Step 4: Stop as soon as it matches. Do not keep “simplifying” past the target.
Do-and-Don’t Rules (to Avoid Memorization Traps)
Do
- Do keep one side unchanged. Write the identity as
Left Side = Right Side, then work only on one side until it becomes the other. - Do annotate each transformation. A short reason prevents illegal moves and makes your logic checkable.
- Do prefer algebra first: factoring, distributing, combining fractions, canceling common factors (only after factoring).
- Do rewrite in
sinandcoswhen stuck. This often turns trig into plain algebra. - Do use common denominators before “fancy” substitutions. Many proofs collapse after a single combine-and-factor step.
Don’t
- Don’t change both sides simultaneously. If you rewrite both sides, you can accidentally move away from equality without noticing.
- Don’t “cancel” across addition/subtraction. You may cancel only common factors, not common terms. For example,
(a+b)/adoes not simplify by cancelinga. - Don’t divide by an expression that could be zero unless you explicitly factor first and understand the domain restrictions. In identity proofs, prefer factoring and cancellation of common factors (which implicitly assumes nonzero factors) and note where expressions are undefined.
- Don’t force an identity. If you are “stuck,” back up and try a different first rewrite (often: common denominator or sin/cos conversion).
Proof 1 (Warm-Up): Quotient/Reciprocal Structure
Prove: (1 - cos x)/sin x = csc x - cot x
Strategy: Work on the left side (single fraction) and aim for a difference of two terms.
Start with: (1 - cos x)/sin x| Step | Expression | Reason |
|---|---|---|
| 1 | (1 - cos x)/sin x = 1/sin x - cos x/sin x | Split the fraction: (A−B)/C = A/C − B/C |
| 2 | 1/sin x - cos x/sin x = csc x - cot x | Rewrite using reciprocal/quotient definitions |
At Step 2 the expression matches the right side exactly, so the proof is complete.
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Proof 2 (Still Basic): Common Denominator Before Anything Else
Prove: tan x + cot x = sec x csc x
Strategy: Work on the left side; combine into one fraction first.
Start with: tan x + cot x| Step | Expression | Reason |
|---|---|---|
| 1 | tan x + cot x = (sin x/cos x) + (cos x/sin x) | Rewrite in sin/cos (quotient identities) |
| 2 | = (sin^2 x + cos^2 x)/(sin x cos x) | Common denominator sin x cos x |
| 3 | = 1/(sin x cos x) | Pythagorean: sin^2 x + cos^2 x = 1 |
| 4 | = (1/sin x)(1/cos x) | Split 1/(ab) into (1/a)(1/b) |
| 5 | = csc x sec x | Reciprocal identities |
Notice how the proof becomes straightforward once everything is in a single fraction.
Proof 3 (Intermediate): Pythagorean Substitution as a Trigger
Prove: (1 - sin^2 x)/cos x = cos x
Strategy: Work on the left side; the pattern 1 - sin^2 is a direct Pythagorean trigger.
Start with: (1 - sin^2 x)/cos x| Step | Expression | Reason |
|---|---|---|
| 1 | (1 - sin^2 x)/cos x = (cos^2 x)/cos x | Pythagorean: 1 - sin^2 x = cos^2 x |
| 2 | = cos x | Cancel common factor cos x (where cos x ≠ 0) |
This is a good example of why you should look for Pythagorean “shapes” before trying more complex identities.
Proof 4 (Intermediate): Factor First, Then Cancel
Prove: (sec x - cos x)/tan x = sin x
Strategy: Work on the left side; rewrite in sin/cos, then factor to create a cancelable factor.
Start with: (sec x - cos x)/tan x| Step | Expression | Reason |
|---|---|---|
| 1 | (sec x - cos x)/tan x = (1/cos x - cos x)/(sin x/cos x) | Rewrite sec and tan in sin/cos |
| 2 | = ((1 - cos^2 x)/cos x)/(sin x/cos x) | Combine terms in the numerator over cos x |
| 3 | = ((1 - cos^2 x)/cos x) · (cos x/sin x) | Divide by a fraction = multiply by its reciprocal |
| 4 | = (1 - cos^2 x)/sin x | Cancel common factor cos x |
| 5 | = (sin^2 x)/sin x | Pythagorean: 1 - cos^2 x = sin^2 x |
| 6 | = sin x | Cancel common factor sin x (where sin x ≠ 0) |
Key habit: you did not “cancel cos” until it was a factor. That prevents illegal cancellation across subtraction.
Proof 5 (Harder): Sum/Difference Pattern Recognition
Prove: (sin(x + y) - sin(x - y)) / (cos(x - y) - cos(x + y)) = cot x
Strategy: Work on the left side; expand using sum/difference formulas, then factor.
Start with: (sin(x + y) - sin(x - y)) / (cos(x - y) - cos(x + y))| Step | Expression | Reason |
|---|---|---|
| 1 | Numerator: sin(x+y) - sin(x-y) = (sin x cos y + cos x sin y) - (sin x cos y - cos x sin y) | Expand sin(x±y) |
| 2 | = sin x cos y + cos x sin y - sin x cos y + cos x sin y | Distribute the minus sign |
| 3 | = 2 cos x sin y | Combine like terms |
| 4 | Denominator: cos(x-y) - cos(x+y) = (cos x cos y + sin x sin y) - (cos x cos y - sin x sin y) | Expand cos(x±y) |
| 5 | = cos x cos y + sin x sin y - cos x cos y + sin x sin y | Distribute the minus sign |
| 6 | = 2 sin x sin y | Combine like terms |
| 7 | Whole fraction = (2 cos x sin y)/(2 sin x sin y) | Substitute simplified numerator and denominator |
| 8 | = cos x/sin x | Cancel common factors 2 and sin y (where sin y ≠ 0) |
| 9 | = cot x | Quotient identity |
Notice the “engine” of the proof: expand, distribute negatives carefully, then factor/cancel.
Proof 6 (Harder): Double-Angle Used as a Finishing Move
Prove: (1 - cos 2x)/sin 2x = tan x
Strategy: Work on the left side; rewrite the double-angle pieces into expressions involving sin x and cos x, then simplify.
Start with: (1 - cos 2x)/sin 2x| Step | Expression | Reason |
|---|---|---|
| 1 | (1 - cos 2x)/sin 2x = (1 - (1 - 2sin^2 x)) / (2sin x cos x) | Use cos 2x = 1 - 2sin^2 x and sin 2x = 2sin x cos x |
| 2 | = (1 - 1 + 2sin^2 x)/(2sin x cos x) | Simplify inside the numerator |
| 3 | = (2sin^2 x)/(2sin x cos x) | Combine like terms |
| 4 | = sin x/cos x | Cancel common factor 2sin x (where sin x ≠ 0) |
| 5 | = tan x | Quotient identity |
Double-angle identities often work best after you’ve reduced the expression to a single fraction where cancellation becomes visible.
Proof 7 (Challenge): Mix of Sum/Difference and Double-Angle
Prove: sin(x + y)sin(x - y) = sin^2 x - sin^2 y
Strategy: Work on the left side; expand both factors, then group and use Pythagorean substitutions. This is a good test of “algebra first.”
Start with: sin(x + y)sin(x - y)| Step | Expression | Reason |
|---|---|---|
| 1 | sin(x+y)sin(x-y) = (sin x cos y + cos x sin y)(sin x cos y - cos x sin y) | Expand sin(x±y) |
| 2 | = (sin x cos y)^2 - (cos x sin y)^2 | Difference of squares: (A+B)(A−B)=A^2−B^2 |
| 3 | = sin^2 x cos^2 y - cos^2 x sin^2 y | Square each factor |
| 4 | = sin^2 x(1 - sin^2 y) - (1 - sin^2 x)sin^2 y | Rewrite cos^2 as 1 - sin^2 (Pythagorean) |
| 5 | = sin^2 x - sin^2 x sin^2 y - sin^2 y + sin^2 x sin^2 y | Distribute |
| 6 | = sin^2 x - sin^2 y | Cancel opposite terms |
This proof shows a common pattern: expand → recognize algebraic structure (difference of squares) → substitute Pythagorean → distribute and simplify.
A Quick Self-Check List While You Write Proof Steps
- Did I keep one side unchanged? If not, restart and pick a side.
- Did I justify each step? Use short reasons like “common denominator,” “Pythagorean,” “difference of squares,” “factor and cancel.”
- Did I cancel only factors? If you canceled terms across +/−, undo it.
- Did I try sin/cos rewriting before advanced identities? If you are stuck, that is usually the next best move.
- Did I stop when it matched? Matching the target is the finish line.
