Additional Mathematics Synthesis Guide 34: Trigonometric Identity Engineering — Addition Formulae, Double Angles, Simplification, Proof and Equation Reduction

Read the target → choose a useful identity → transform one side or the equation → preserve zero cases → solve completely → verify.

1. Addition formulae

• sin(A+B)=sinA cosB+cosA sinB
• sin(A−B)=sinA cosB−cosA sinB
• cos(A+B)=cosA cosB−sinA sinB
• cos(A−B)=cosA cosB+sinA sinB
• tan(A±B)=(tanA±tanB)/(1∓tanA tanB), where defined.

These formulas let one compound angle be expanded into simpler angles, or let a structured expression be compressed back into one compound angle.

2. Double-angle identities

Set A=B in the addition formulae:

• sin2A=2sinA cosA
• cos2A=cos²A−sin²A
• cos2A=1−2sin²A
• cos2A=2cos²A−1
• tan2A=2tanA/(1−tan²A), where defined.

The three cosine forms are equivalent. Choose the one that reduces the number of trig functions in the problem.

3. Exact value from a compound angle

Find sin75° exactly:

sin(45°+30°)=sin45°cos30°+cos45°sin30°

=(√2/2)(√3/2)+(√2/2)(1/2)=(√6+√2)/4.

The exact surd form should be kept unless approximation is requested.

4. Exact cosine value

Find cos15° exactly:

cos(45°−30°)=cos45°cos30°+sin45°sin30°

=(√6+√2)/4.

The same exact expression appears because sin75°=cos15°.

5. A proof should have a direction

To prove an identity, usually begin with the more complicated side and transform it into the simpler side. Avoid writing the statement to be proved and then performing the same operation simultaneously on both sides as though equality were already established.

The proof is a chain of known equalities from one expression to the other.

6. Worked identity proof

Prove

(1−cos2x)/sin2x=tanx

where the expressions are defined.

Use 1−cos2x=2sin²x and sin2x=2sinx cosx:

(2sin²x)/(2sinx cosx)=sinx/cosx=tanx.

The cancellation assumes the original denominator is non-zero. The identity is interpreted only where both sides are defined.

7. Identity domain matters

An identity such as secx=1/cosx is not a statement about points where cosx=0. Both sides are undefined there.

Identity proof does not erase domain restrictions. A correct simplification is always attached to the domain on which the expressions exist.

8. Reduce an equation to one trig function

Solve structurally:

cos2x=sinx.

Use cos2x=1−2sin²x:

1−2sin²x=sinx.

Let u=sinx:

2u²+u−1=0=(2u−1)(u+1).

So sinx=1/2 or −1. Guide 28 then supplies the interval-complete angle solutions.

9. Choose among equivalent cosine-double-angle forms

If an equation contains cos2x and sinx, use 1−2sin²x. If it contains cos2x and cosx, use 2cos²x−1. If both sin²x and cos²x are already present, cos²x−sin²x may be cheapest.

Identity choice is method selection, not formula dumping.

10. Do not divide away a solution branch

Consider

sin2x=sinx.

Use sin2x=2sinx cosx:

2sinx cosx=sinx

sinx(2cosx−1)=0.

Therefore sinx=0 or cosx=1/2. Dividing by sinx would lose the entire sinx=0 branch.

11. Compound-angle compression

An expression such as

sinx cos30°+cosx sin30°

is exactly

sin(x+30°).

Recognition can shorten an equation dramatically.

12. Worked compression equation

Solve

√3 sinx+cosx=1.

One route is R-form. Since amplitude is 2, write

√3 sinx+cosx=2sin(x+30°).

Then solve sin(x+30°)=1/2 over the transformed interval. Guide 17 develops R-form and Guide 28 interval completion.

13. Tangent addition formula and restrictions

For tan(A+B), the denominator 1−tanA tanB must be non-zero, and tanA,tanB must themselves be defined.

When using tangent identities in equations, retain these restrictions rather than treating a rational formula as globally valid.

14. Deriving tan75° exactly

Using tan(45°+30°):

tan75°=(1+1/√3)/(1−1/√3).

Multiply numerator and denominator by √3:

(√3+1)/(√3−1)=2+√3.

Exact surd algebra and trigonometric identities meet here directly.

15. Factor before converting

For an equation such as

2sinx cosx−sinx=0,

factor first:

sinx(2cosx−1)=0.

Converting immediately to sin2x−sinx=0 is valid, but factorisation is already closer to the solution branches.

16. Convert squares when they obstruct solving

If an equation contains sin²x and cosx, use sin²x=1−cos²x to obtain one function. If it contains sin²x−cos²x, use −cos2x if that simplifies the angle structure.

There is no single best identity independent of the target.

17. Proof by target matching

Suppose the target contains tanx. Since tanx=sinx/cosx, a proof expression containing sin and cos may be transformed toward a quotient with those factors.

Suppose the target contains sec²x. Then the Pythagorean identity 1+tan²x=sec²x is likely relevant.

The target gives information about which representation is useful.

18. Verify identities numerically — but do not prove them numerically

Testing x=0.4 radians can catch a sign error in a proposed identity. But agreement at several numerical values is not a proof of equality for all allowed x.

Use numerical checks for debugging; use algebraic identities for proof.

19. Exact angles and calculator state

Compound-angle identities may be written in degrees or radians, but calculus-linked trigonometric work conventionally uses radians. Keep the angle unit consistent with the question and calculator mode.

Guide 28 develops principal values and calculator-state control.

20. Identity chains should preserve meaning

A long derivation should make every equality justified. If a step multiplies by a denominator, divides by a variable expression, squares both sides or takes an inverse function, note whether information may be lost or additional candidates created.

Identity transformations are usually reversible on their domains; equation operations need case control.

21. Common failure patterns

• Choosing an identity because it is remembered rather than because it reduces the target.
• Using the wrong sign in sin(A−B) or cos(A+B).
• Using one cosine-double-angle form when another would reduce the equation to one trig function.
• Dividing by sinx or cosx and losing zero branches.
• Treating a few numerical checks as proof.
• Ignoring domain restrictions in tangent, secant or quotient identities.
• Expanding both sides of an identity until the structure becomes harder to see.
• Finding a reference equation but failing to complete all interval solutions.

22. A reliable identity-engineering routine

1. Read the target and count how many trig functions and angle forms are present.
2. Choose an identity that reduces that structural count.
3. For proof, transform the more complicated side toward the target.
4. For equations, preserve zero-factor branches before division or cancellation.
5. Reduce to one trig function where possible.
6. Carry all domain and denominator restrictions.
7. Use exact values until approximation is requested.
8. Complete the solution over the full interval and verify candidates.

23. Practice set

1. Find sin75° exactly.
2. Find cos15° exactly.
3. Find tan75° exactly.
4. Expand sin(A−B).
5. Expand cos(A+B).
6. Write sin2x in product form.
7. Write cos2x entirely in sinx.
8. Write cos2x entirely in cosx.
9. Prove (1−cos2x)/sin2x=tanx where defined.
10. Reduce cos2x=sinx to a quadratic in sinx.
11. Factor sin2x−sinx.
12. Explain why dividing by sinx can lose solutions.
13. Compress sinx cos30°+cosx sin30°.
14. Compress cosx cos20°−sinx sin20°.
15. Which cosine-double-angle form is best if an equation contains cos2x and cosx?
16. Why is numerical checking not an identity proof?
17. State one restriction in tan(A+B).
18. Show that sin²x=(1−cos2x)/2.
19. Show that cos²x=(1+cos2x)/2.
20. Why should the target guide identity selection?

Answers

1. (√6+√2)/4.
2. (√6+√2)/4.
3. 2+√3.
4. sinA cosB−cosA sinB.
5. cosA cosB−sinA sinB.
6. 2sinx cosx.
7. 1−2sin²x.
8. 2cos²x−1.
9. Use 1−cos2x=2sin²x and sin2x=2sinx cosx; simplify to tanx on the common domain.
10. 1−2sin²x=sinx ⇒ 2sin²x+sinx−1=0.
11. sinx(2cosx−1).
12. Because solutions with sinx=0 are excluded by the division.
13. sin(x+30°).
14. cos(x+20°).
15. 2cos²x−1.
16. Finite sample agreement does not establish equality for every allowed angle.
17. 1−tanA tanB≠0, and tanA,tanB must be defined.
18. From cos2x=1−2sin²x, rearrange.
19. From cos2x=2cos²x−1, rearrange.
20. Because the most useful identity is the one that transforms the current expression toward the required final representation.

24. What mastery looks like

Mastery means the learner selects identities by structural purpose, can move between compound-angle and expanded forms, derives exact values, proves identities through justified equalities, reduces equations to one trig function and protects solution branches when a transformation is not reversible.

The transfer test is to provide an unfamiliar identity or mixed equation without naming the formula. If the learner can explain why a particular representation reduces the problem and can preserve all domain and interval conditions, trigonometric identity work has become engineered rather than memorised.

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Continue through Batch 09

• Guide 33: Exponential and Logarithmic Functions — Laws, Change of Base, Inverse Graphs, Equations and Models
• Guide 35: Kinematics from Calculus — Displacement, Velocity, Acceleration, Direction Changes and Total Distance
• Guide 36: Graph-to-Equation Reconstruction — Quadratics, Exponentials, Trigonometric Functions and Parameter Recovery
• Additional Mathematics Directory
• BTT Mathematics Hub