Sum and difference formulas — sin(A±B), cos(A±B), tan(A±B) — proofs

3.1.12 · Maths › Advanced Trigonometry

1. Hum kya prove kar rahe hain

Sign gymnastics note karo: cosine mein signs flip hote hain (bahar plus → andar minus), sine mein signs match karte hain. Yahi pattern poora memory game hai.

2. Scratch se derivation — unit-circle / distance method

Hum pehle == $cos (A - B)$ == prove karte hain, kyunki baaki sab isi se nikal aata hai.

Figure — Sum and difference formulas — sin(A±B), cos(A±B), tan(A±B) — proofs

Step 1 — Points rakho

Unit circle par:

$P = (cos A, sin A)$
$Q = (cos B, sin B)$

Ye step kyun? Definition se, unit circle par angle $θ$ par ek point ke coordinates $(cos θ, sin θ)$ hote hain. Isse angles geometry ban jaate hain.

Step 2 — Chord length via coordinates (distance formula)

$P Q^{2} = (cos A - cos B)^{2} + (sin A - sin B)^{2}$ Expand karo: $= cos^{2} A - 2 cos A cos B + cos^{2} B + sin^{2} A - 2 sin A sin B + sin^{2} B$ $cos^{2} θ + sin^{2} θ = 1$ use karke group karo: $= 1 (cos^{2} A + sin^{2} A) + 1 (cos^{2} B + sin^{2} B) - 2 (cos A cos B + sin A sin B)$ $P Q^{2} = 2 - 2 (cos A cos B + sin A sin B)$

Ye step kyun? Pythagorean identity chaar squared terms ko ek clean constant mein collapse kar deti hai, sirf cross-terms bachte hain — aur humara formula usi ke baare mein hai.

Step 3 — Chord length via unke beech ka angle

$Q$ se $P$ tak ka angle $(A - B)$ hai. Poori picture rotate karo taaki $Q$ angle $0$ par aa jaaye, yaani $(1, 0)$ par, aur $P$ angle $(A - B)$ par aa jaaye, yaani $(cos (A - B), sin (A - B))$ par. Rotation se distances nahi badlti, isliye: $P Q^{2} = (cos (A - B) - 1)^{2} + (sin (A - B) - 0)^{2}$ $= cos^{2} (A - B) - 2 cos (A - B) + 1 + sin^{2} (A - B)$ $= 1 - 2 cos (A - B) + 1 = 2 - 2 cos (A - B)$

Ye step kyun? Distance rotation ke under invariant hai, isliye wahi chord geometry simplify karne ke baad measure ki ja sakti hai — ab sirf $(A - B)$ bachi hai.

Step 4 — Dono expressions ko barabar karo

$2 - 2 cos (A - B) = 2 - 2 (cos A cos B + sin A sin B)$ $∴ cos (A - B) = cos A cos B + sin A sin B ✓$

3. Baaki paanch free mein lena

4. Worked examples

5. Forecast-then-Verify

Recall Expand karne se pehle predict karo:

$cos (A + B) + cos (A - B)$ kya hai? Ek product ka andaza lagao... Expand karo: $(cos A cos B - sin A sin B) + (cos A cos B + sin A sin B) = 2 cos A cos B$ $sin sin$ terms cancel ho jaate hain → yahi product-to-sum identity hai. In formulas ka sum aadhi terms cancel kar deta hai; yahi saare product-to-sum results ke peeche ka engine hai.

6. Common mistakes (Steel-manned)

7. Flashcards

Pehle kaun si identity prove hoti hai, aur kis method se?

cos (A - B) = cos A cos B + sin A sin B

, unit circle par chord/distance formula se.

sin (A + B)

batao.

sin A cos B + cos A sin B

cos (A + B)

batao.

cos A cos B - sin A sin B

(signs flip hote hain).

tan (A - B)

batao.

\frac{tan A - tan B}{1 + tan A tan B}

cos (A - B)

cos (A + B)

kaise nikalte hain?

B \to - B

replace karo aur

cos

even,

sin

odd use karo.

Cosine formula se sine formula kaise banate hain?

Cofunction:

sin θ = cos (\frac{π}{2} - θ)

Tangent proof mein

cos A cos B

se divide kyun karte hain?

Taaki har

sin / cos

ratio ek

tan

ban jaaye.

Exact

sin 75°

\frac{6 + 2}{4}

Ek-line reason ki

sin (A + B) \neq = sin A + sin B

kyun hai?

Sine rotation encode karta hai, linear scaling nahi.

Recall Feynman: 12-saal ke bachche ko explain karo

Ek ghadi ki sooyi imagine karo jo ghoom rahi hai. Use angle $A$ se ghumao, phir thoda aur $B$ se ghumao — yahi jagah milti hai jahan seedha $A + B$ se ghumane par milti. Sine aur cosine bas sooyi ki nok ki "height" aur "sideways" position hain. Ye formulas woh recipe hain jo bataate hain: dono turns ke baad nayi height/sideways jaanne ke liye, har turn ki heights aur sideways positions ko aapas mein mix karo. Isliye answer sirf add karna nahi hai — do turns aapas mein ghul-mil jaate hain, kuch parts add hote hain, kuch subtract.

Connections

Unit Circle and Trig Definitions — coordinates $(cos θ, sin θ)$ se sab shuru hota hai.
Double Angle Formulas — inmein $B = A$ rakho.
Product-to-Sum and Sum-to-Product — in identities ko add/subtract karo.
Pythagorean Identity — Step 2 mein collapse trick.
Derivatives of Trig Functions — $\frac{d}{d x} sin x$ ki limit proof mein $sin (x + h)$ expansion use hoti hai.
Cofunction Identities — cosine se sine proofs tak ka bridge.