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Trigonometric Identities Made Simple: Choosing the Right Identity From the Form

Capítulo 8

Estimated reading time: 7 minutes

Identity Choice Is a Form Problem

Most trig simplifications fail not because the algebra is hard, but because the first identity choice increases complexity. A practical way to choose is to decide what you want the expression to become (the target form), then look for visible “triggers” that suggest a specific identity. Your goal is to reduce the number of distinct trig functions (sin, cos, tan, sec, etc.) and/or reduce the number of distinct angles (x, 2x, x/2, x±y).

Step 1: Name the Target Form

Before touching the expression, classify the goal:

Target A: a single trig function (e.g., “make it all sin and cos,” or “make it all tan”).
Target B: a constant (often 0, 1, or a simple number).
Target C: a simplified ratio (a reduced fraction, often in sin/cos or tan/sec form).

Then apply: choose the identity that reduces the number of distinct functions or angles fastest.

Step 2: Scan for Triggers (What the Form Is “Asking For”)

Trigger you see	Likely best move	Why it helps
Squares like `sin^2 x`, `cos^2 x`, or `1 − sin^2 x`	Use a Pythagorean variant to swap `1 − sin^2 x` ↔ `cos^2 x` or `1 − cos^2 x` ↔ `sin^2 x`	Turns “1 ± trig^2” into a clean square; often cancels with a denominator.
`1 + tan^2 x` or `1 + cot^2 x`	Convert to `sec^2 x` or `csc^2 x`	Immediately reduces the number of terms; often collapses to a single square.
Mixed `sec` and `tan` (or `csc` and `cot`)	Use `sec^2 x − tan^2 x = 1` (or `csc^2 x − cot^2 x = 1`) when you can form it	These pairs are “designed” to become a constant.
Different angles present (e.g., `x` and `2x`)	Convert everything to one angle (often `x`) using a double-angle form that matches the expression	Angle mismatch is a common reason simplification stalls.
Sum/difference angles like `sin(x±y)`, `cos(x±y)`	Expand only if it helps cancellation or matches other factors; otherwise keep compact	Expanding can double term count; do it only with a purpose.
Rational expression in `sin` and `cos` that “looks like” a tangent	Consider rewriting as `tan x` or `cot x` if it reduces complexity	Sometimes a ratio is already a quotient identity in disguise.

Efficiency Rule: Reduce Variety First

When two paths are possible, prefer the one that:

reduces the number of distinct trig functions sooner (e.g., from 3 functions down to 1),
reduces the number of distinct angles sooner (e.g., eliminate 2x early),
avoids expanding products unless it creates immediate cancellation.

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Function variety: how many of {sin, cos, tan, sec, csc, cot} appear?
Angle variety: how many different angles appear (x, 2x, x/2, x±y)?

Your first identity choice should usually decrease at least one of these counts.

Worked Comparisons: Two Correct Paths, One Shorter

Comparison 1: Constant Target (Spot the “sec/tan trigger”)

Simplify: sec^2 x − tan^2 x

Target form: constant.

Trigger: mixed sec/tan squares in the exact pattern.

Efficient path: use the identity directly.

sec^2 x − tan^2 x = 1

Longer path (works but inefficient): rewrite in sin/cos first.

sec^2 x − tan^2 x = (1/cos^2 x) − (sin^2 x/cos^2 x) = (1 − sin^2 x)/cos^2 x = cos^2 x/cos^2 x = 1

Efficiency choice: if the expression already matches a “constant-maker” pattern, do not convert to sin/cos first.

Comparison 2: Simplified Ratio Target (Cancel a square vs expand)

Simplify: (1 − sin^2 x)/cos x

Target form: simplified ratio (single trig function if possible).

Trigger: 1 − sin^2 x.

Efficient path: convert the numerator to a square that cancels.

(1 − sin^2 x)/cos x = cos^2 x/cos x = cos x

Alternative path (works but longer): rewrite everything in terms of tan/sec or expand into separate fractions (adds steps and rarely helps here).

Efficiency choice: when you see 1 ± trig^2, try to turn it into a single square that cancels with a denominator.

Comparison 3: Angle Variety Problem (Eliminate 2x early)

Simplify: (1 − cos 2x)/(sin x)

Target form: simplified ratio, ideally a single function of x.

Trigger: a 2x angle mixed with x.

Path A (efficient): choose a double-angle form that produces sin x to cancel.

1 − cos 2x = 2 sin^2 x

(1 − cos 2x)/sin x = (2 sin^2 x)/sin x = 2 sin x

Path B (works but longer): use cos 2x = 1 − 2 sin^2 x, then simplify (still okay), or convert cos 2x to cos^2 x − sin^2 x (often increases term count).

Efficiency choice: pick the 2x identity that creates immediate cancellation with what you already have.

Comparison 4: To Expand or Not (Sum/Difference trigger)

Simplify: sin(x + y)sin(x − y)

Target form: simplified expression (often in squares).

Trigger: product of sum/difference angles.

Two possible paths:

Path A (expand both sines): leads to four terms and then regrouping.
Path B (use a product-to-sum identity): collapses immediately.

Efficient path (B):

sin(x + y)sin(x − y) = (1/2)[cos((x + y) − (x − y)) − cos((x + y) + (x − y))]

= (1/2)[cos(2y) − cos(2x)]

Efficiency choice: if expanding creates many terms, look for a product-to-sum or sum-to-product shortcut that reduces term count.

A Practical Decision Framework (Checklist)

1) Identify the target

If the expression resembles a known constant pattern (like a difference of squares in sec/tan), aim for constant.
If it’s a rational expression, aim for a simplified ratio with cancellations.
If the problem statement asks “write in terms of …”, aim for a single function family (often sin/cos).

2) Reduce angle variety

If both x and 2x appear, pick a 2x identity that creates cancellation with existing factors.
If x±y appears, expand only when it matches other pieces for cancellation; otherwise consider product-to-sum.

3) Reduce function variety

If you see 1 ± trig^2, convert it to a square (often cancels).
If you see mixed reciprocal functions (sec, csc), decide whether converting to sin/cos will reduce variety or increase it.
If you see tan with sec, try to form sec^2 − tan^2 or 1 + tan^2 patterns.

Mixed Practice: Justify the Identity Choice First

For each problem, do two lines before simplifying: (1) target form, (2) trigger + chosen identity. Then simplify.

Set A: Spot the Trigger

1. (1 + tan^2 x)/(sec^2 x)
2. (1 − cos^2 x)/sin x
3. sec^2 x − 1
4. (1 − cos 2x)/(1 + cos 2x)
5. (sin^2 x + cos^2 x)tan x

Set B: Two Paths Exist—Choose the Shorter and Say Why

6. (sec x − tan x)(sec x + tan x)
7. (1 − sin^2 x)/(1 − sin x)
8. sin 2x/(1 + cos 2x)
9. (1 + cos 2x)/2
10. (cos(x + y) + cos(x − y))

Set C: Angle Variety Control (Unify the Angle)

11. (1 − cos 2x)/sin 2x
12. (sin 2x)/(sin x)
13. (1 + cos 2x)/(sin x cos x)
14. cos 2x + 2 sin^2 x
15. (sin(x + y) − sin(x − y))/cos y

Instructor Key (Identity-Choice Justification Only)

#	Target form	Trigger	Identity choice (first move)
1	constant	`1 + tan^2 x`	replace with `sec^2 x` to cancel
2	single function	`1 − cos^2 x`	replace with `sin^2 x` to cancel with `sin x`
3	single function	`sec^2 x − 1`	replace with `tan^2 x`
4	simplified ratio	`1 ± cos 2x`	use forms that turn into `sin^2 x` and `cos^2 x` to reduce to a square ratio
5	single function	`sin^2 x + cos^2 x`	replace with `1`
6	constant	conjugates	multiply as difference of squares, then use `sec^2 − tan^2 = 1`
7	simplified ratio	`1 − sin^2 x` and `1 − sin x`	convert numerator to `(1 − sin x)(1 + sin x)` via `1 − sin^2 x` factorization
8	single function	`sin 2x` with `1 + cos 2x`	use `sin 2x = 2 sin x cos x` and `1 + cos 2x = 2 cos^2 x` for cancellation
9	single function	`(1 + cos 2x)/2`	use the half-angle connection to rewrite as a square in `x`
10	simplified expression	sum of cosines	use sum-to-product to reduce term count
11	single function	`1 − cos 2x` over `sin 2x`	rewrite numerator as `2 sin^2 x` and denominator as `2 sin x cos x`
12	simplified ratio	`sin 2x` with `sin x`	rewrite `sin 2x` to cancel `sin x`
13	single function	`1 + cos 2x` and product `sin x cos x`	rewrite `1 + cos 2x` as `2 cos^2 x` to cancel a `cos x`
14	constant/simplified	`cos 2x` plus `sin^2 x`	choose a `cos 2x` form involving `sin^2 x` to combine like terms
15	single function	difference of sines	use sum-to-product to factor out `cos y` for cancellation

Now answer the exercise about the content:

When simplifying (1 − cos 2x)/sin x, which first identity choice is most efficient and why?

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The expression mixes 2x and x, and the denominator has sin x. Using 1 - cos 2x = 2 sin^2 x creates an immediate cancellation with sin x, quickly reducing angle and term complexity.