NDA Maths · Trigonometric Identities

Product-to-Sum & Sum-to-Product

The two conversion families — turn a product of sines/cosines into a sum, or a sum into a product — plus the telescoping product chains and the conditional identities for A + B + C = 90° or 180°.

All Trigonometric Identities notes NDA Maths strategy

Why this matters

These conversions are what make otherwise-intractable products like 8 cos 10° cos 20° cos 40° or sums like cos 48° − cos 12° collapse to a clean value. Choosing the right direction (product→sum vs sum→product) is the whole decision.

Concept 1 of 4

Product-to-sum formulas

Intuition

Turn a product of two sines/cosines into a sum or difference — derived directly by adding and subtracting the compound-angle formulas. Use this when you have a product and want it to telescope or cancel.

Definition

$2\sin A\cos B=\sin(A+B)+\sin(A-B)$ .
$2\cos A\sin B=\sin(A+B)-\sin(A-B)$ .
$2\cos A\cos B=\cos(A+B)+\cos(A-B)$ .
$2\sin A\sin B=\cos(A-B)-\cos(A+B)$ .

The four product-to-sum identities

2\sin A\cos B=\sin(A+B)+\sin(A-B),\quad 2\cos A\cos B=\cos(A+B)+\cos(A-B)

Worked example

Evaluate

2\sin 75°\cos 15°

$2\sin A\cos B=\sin(A+B)+\sin(A-B)$ with $A=75°,B=15°$ .
$=\sin 90°+\sin 60°=1+\tfrac{\sqrt3}{2}$ .

Answer:

1+\tfrac{\sqrt3}{2}

Practice this conceptself-check · 4 quick reps

Try it yourself

Evaluate

2\cos 75°\cos 15°

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$2\sin A\cos B=?$
2.
$2\sin A\sin B=?$
3.
$2\cos A\cos B=?$
4.
When do you reach for product-to-sum?

From the bank · past-year question

Example 1Trigonometric IdentitiesMODERATE

What is

\dfrac{1}{\sin 10°} - \dfrac{\sqrt{3}}{\cos 10°}

equal to?

[Q40 · Apr · 2017]

Drill 7 more on product-to-sum formulas

Concept 2 of 4

Sum-to-product formulas

Intuition

The reverse direction: a sum or difference of two sines/cosines becomes a product. Use this when you want a common factor to cancel or a ratio to simplify to a single tangent.

Definition

$\sin C+\sin D=2\sin\tfrac{C+D}{2}\cos\tfrac{C-D}{2}$ ; $\;\sin C-\sin D=2\cos\tfrac{C+D}{2}\sin\tfrac{C-D}{2}$ .
$\cos C+\cos D=2\cos\tfrac{C+D}{2}\cos\tfrac{C-D}{2}$ ; $\;\cos C-\cos D=-2\sin\tfrac{C+D}{2}\sin\tfrac{C-D}{2}$ .

Corollary: $\dfrac{\sin C+\sin D}{\cos C+\cos D}=\tan\tfrac{C+D}{2}$ .

Sum-to-product

\sin C+\sin D=2\sin\tfrac{C+D}{2}\cos\tfrac{C-D}{2},\quad \cos C-\cos D=-2\sin\tfrac{C+D}{2}\sin\tfrac{C-D}{2}

Worked example

Simplify

\dfrac{\sin 5x-\sin 3x}{\cos 5x+\cos 3x}

Numerator $=2\cos 4x\sin x$ ; denominator $=2\cos 4x\cos x$ .
Ratio $=\dfrac{\sin x}{\cos x}=\tan x$ .

Answer:

\tan x

Practice this conceptself-check · 4 quick reps

Try it yourself

Evaluate

\cos 48°-\cos 12°

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$\sin C+\sin D=?$
2.
$\cos C-\cos D=?$
3.
$\dfrac{\sin C+\sin D}{\cos C+\cos D}=?$
4.
$\sin 5x-\sin 3x=?$

From the bank · past-year question

Example 2Trigonometric IdentitiesMODERATE

What is the value of

\cos 48°-\cos 12°

[Q28 · Apr · 2020]

Drill 10 more on sum-to-product formulas

Concept 3 of 4

Telescoping products of cosines/sines

Intuition

A chain like cos 10° cos 20° cos 40° collapses by repeatedly using

2\sin\theta\cos\theta=\sin 2\theta

: introduce a sine, and each cosine doubles the angle until the product telescopes.

Definition

Multiply and divide by $2\sin(\text{smallest angle})$ , then apply $2\sin\theta\cos\theta=\sin 2\theta$ repeatedly. General result: $\cos\theta\cos 2\theta\cos 4\theta\cdots\cos 2^{n-1}\theta=\dfrac{\sin 2^n\theta}{2^n\sin\theta}$ . Triple products like $\sin\theta\sin(60°-\theta)\sin(60°+\theta)=\tfrac14\sin 3\theta$ also appear.

Worked example

Evaluate

\cos 20°\cos 40°\cos 80°

Multiply and divide by $2\sin 20°$ : $\dfrac{2\sin 20°\cos 20°\cos 40°\cos 80°}{2\sin 20°}=\dfrac{\sin 40°\cos 40°\cos 80°}{2\sin 20°}$ .
Repeat: $\to\dfrac{\sin 160°}{8\sin 20°}=\dfrac{\sin 20°}{8\sin 20°}=\tfrac18$ .

Answer:

\tfrac18

Practice this conceptself-check · 4 quick reps

Try it yourself

Evaluate

\cos 36°\cos 72°

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Identity to telescope a cosine chain?
2.
$\cos\theta\cos 2\theta\cos 4\theta=?$
3.
$\sin\theta\sin(60°-\theta)\sin(60°+\theta)=?$
4.
First move for $\cos 20°\cos 40°\cos 80°$ ?

From the bank · past-year question

Example 3Trigonometric IdentitiesHARD

What is the value of

8\cos 10°\cdot\cos 20°\cdot\cos 40°

[Q27 · Apr · 2020]

Drill 5 more on telescoping products of cosines/sines

Concept 4 of 4

Conditional identities (A + B + C = 90° or 180°)

Intuition

When three angles sum to 90° or 180°, special identities kick in — the staple results for triangle-angle problems. Recognising the angle-sum condition is the trigger.

Definition

** $A+B+C=180°$ :** $\tan A+\tan B+\tan C=\tan A\tan B\tan C$ ; $\;\sin 2A+\sin 2B+\sin 2C=4\sin A\sin B\sin C$ .
** $A+B+C=90°$ :** $\tan A\tan B+\tan B\tan C+\tan C\tan A=1$ ; $\;\cot A+\cot B+\cot C=\cot A\cot B\cot C$ .

The two signature conditional identities

A+B+C=\pi:\ \tan A+\tan B+\tan C=\tan A\tan B\tan C

Worked example

A+B+C=180°

and

\tan A=1,\tan B=2

, find

\tan C

$\tan A+\tan B+\tan C=\tan A\tan B\tan C\Rightarrow 1+2+\tan C=2\tan C$ .
$3=\tan C$ .

Answer:

\tan C=3

Practice this conceptself-check · 4 quick reps

Try it yourself

A+B+C=180°

, simplify

\sin 2A+\sin 2B+\sin 2C

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$A+B+C=180°$ : $\tan A+\tan B+\tan C=?$
2.
$A+B+C=90°$ : $\sum\tan A\tan B=?$
3.
$A+B+C=180°$ : $\sin 2A+\sin 2B+\sin 2C=?$
4.
What triggers these identities?

From the bank · past-year question

Example 4Trigonometric IdentitiesMODERATE

What is

\tan25°\tan15° + \tan15°\tan50° + \tan25°\tan50°

equal to ?

[Q70 · Sep · 2019]

Drill 1 more on conditional identities (a + b + c = 90° or 180°)

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Product-to-sum formulas
The four product-to-sum identities
$2\sin A\cos B=\sin(A+B)+\sin(A-B),\quad 2\cos A\cos B=\cos(A+B)+\cos(A-B)$
Sum-to-product formulas
Sum-to-product
$\sin C+\sin D=2\sin\tfrac{C+D}{2}\cos\tfrac{C-D}{2},\quad \cos C-\cos D=-2\sin\tfrac{C+D}{2}\sin\tfrac{C-D}{2}$
Conditional identities (A + B + C = 90° or 180°)
The two signature conditional identities
$A+B+C=\pi:\ \tan A+\tan B+\tan C=\tan A\tan B\tan C$

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Trigonometric IdentitiesHARD

Let

p=\cos\frac{\pi}{5}\cos\frac{2\pi}{5}

and

q=\cos\frac{4\pi}{5}\cos\frac{8\pi}{5}

What is the value of

pq

[Q74 · Sep · 2023]

Example 2Trigonometric IdentitiesMODERATE

Consider the following for the items that follow: Let

p = \sin 35°,\ q = \sin 25°

and

r = \sin(-95°)

What is

(p + q + r)

equal to?

[Q31 · Apr · 2025]

Example 3Trigonometric IdentitiesHARD

What is

\cot 2x \cot 4x - \cot 4x \cot 6x - \cot 6x \cot 2x

equal to?

[Q15 · Apr · 2021]

Example 4Trigonometric IdentitiesHARD

A + B + C = 180°

, then what is

\sin 2A - \sin 2B - \sin 2C

equal to?

[Q38 · Sep · 2018]

Example 5Trigonometric IdentitiesHARD

What is the value of

\cos\!\left(\frac{5\pi}{17}\right)+\cos\!\left(\frac{7\pi}{17}\right)+2\cos\!\left(\frac{11\pi}{17}\right)\cos\!\left(\frac{\pi}{17}\right)

[Q47 · Sep · 2022]

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