Identities & Complementary Angles

🎯 Learning Objective

Use sin²θ + cos²θ = 1 and the 90° − θ relations to simplify expressions without tables.

💡 Concept

• sin²θ + cos²θ = 1 — the master identity
• 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ
• Complementary pairs: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, sec(90° − θ) = cosec θ
• sec²θ − tan²θ = 1 factorises: (sec θ − tan θ)(sec θ + tan θ) = 1

🧮 Key Formulas

sin²θ + cos²θ = 1

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1 + tan²θ = sec²θ

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sin(90° − θ) = cos θ

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(sec θ − tan θ)(sec θ + tan θ) = 1

✏️ Easy Example

Q. If sin θ = 3/5, find cos θ. (θ is acute)

1. cos²θ = 1 − sin²θ
2. = 1 − 9/25 = 16/25
3. cos θ = 4/5

Answer: 4/5

🇮🇳 Real-Life Example

Lean a ladder any way you like — (horizontal reach ÷ length)² + (height reached ÷ length)² always equals 1. That is sin² + cos² = 1 standing in your courtyard.

📝 Exam-Level Example

Q. Find the value of sin 25° / cos 65°.

1. cos 65° = sin(90° − 65°) = sin 25°
2. sin 25° / sin 25°

Answer: 1

📝 Exam-Level Example

Q. If sec θ − tan θ = 1/3, find sec θ + tan θ.

1. (sec θ − tan θ)(sec θ + tan θ) = 1
2. sec θ + tan θ = 1 ÷ (1/3)

Answer: 3

🪄 Memory Trick

Angles adding to 90°? Convert one ratio into its co-ratio and cancel: sin 25°/cos 65° = 1 with zero table work.

⚠️ Common Mistakes

• ❌ sin²θ means (sin θ)², not sin(θ²)
• ❌ Applying the complementary rule when the two angles do not add to 90°

🏆 Exam Tips

• ✅ sec θ − tan θ = k given → sec θ + tan θ = 1/k, instantly
• ✅ Hunt for sin²θ + cos²θ hidden inside long expressions and replace it with 1

📌 Summary

• sin² + cos² = 1 family of three identities
• 90° − θ swaps sin↔cos, tan↔cot, sec↔cosec
• (sec − tan)(sec + tan) = 1
• Angles summing to 90° → convert and cancel