Question
Prove that sin²θ + cos²θ = 1.
Also prove the related identities:
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
Solution — Step by Step
This proof is fundamental. Everything else in trigonometric identities builds on this. Let's prove it properly.
Step 1: Set up a right-angled triangle.
Consider a right-angled triangle ABC where angle C = 90° and angle A = θ.
Label the sides:
- BC = opposite side (let its length = a)
- AC = adjacent side (let its length = b)
- AB = hypotenuse (let its length = c)
Step 2: Write the trig ratios.
sin θ = opposite/hypotenuse = a/c
cos θ = adjacent/hypotenuse = b/c
Step 3: Square and add.
sin²θ + cos²θ = (a/c)² + (b/c)² = a²/c² + b²/c² = (a² + b²) / c²
Step 4: Apply Pythagoras' theorem.
In a right-angled triangle: a² + b² = c² (Pythagoras' theorem)
So: sin²θ + cos²θ = c²/c² = 1 ✓
Proof Complete
Since a² + b² = c² (Pythagorean theorem),
sin²θ + cos²θ = (a² + b²)/c² = c²/c² = 1
Step 5: Prove 1 + tan²θ = sec²θ.
Start with the proven identity: sin²θ + cos²θ = 1
Divide every term by cos²θ:
sin²θ/cos²θ + cos²θ/cos²θ = 1/cos²θ
tan²θ + 1 = sec²θ
Rewriting: 1 + tan²θ = sec²θ ✓
Step 6: Prove 1 + cot²θ = cosec²θ.
Start again with: sin²θ + cos²θ = 1
Divide every term by sin²θ:
sin²θ/sin²θ + cos²θ/sin²θ = 1/sin²θ
1 + cot²θ = cosec²θ ✓
Why This Works
The entire proof rests on Pythagoras' theorem applied to the sides of a right triangle. The trig ratios are just names for the ratios of specific sides. When you square them and add, you naturally reconstruct the Pythagorean relationship a² + b² = c² — hence the name "Pythagorean identities."
Unit Circle Proof (Alternative)
The unit circle approach works for all angles, not just acute angles in a right triangle.
On a unit circle (radius = 1), any point on the circle has coordinates (cos θ, sin θ). Since the point is on the circle of radius 1:
x² + y² = 1
So: cos²θ + sin²θ = 1 ✓
This proof is more general — it works for θ = 120°, 270°, or any angle.
💡 Expert Tip
To remember which form of the Pythagorean identity to use in a problem: if you see sec or tan in the expression, use 1 + tan²θ = sec²θ. If you see cosec or cot, use 1 + cot²θ = cosec²θ. The basic form handles everything with sin and cos directly.
Common Mistake
⚠️ Common Mistake
Mistake: Writing the proof by starting from the identity and manipulating both sides simultaneously.
A proof must start from known facts and arrive at the result. Never start by writing "sin²θ + cos²θ = 1 ∴ sin²θ = 1 − cos²θ..." — that assumes what you're trying to prove. Start from the definition of sin and cos (sides of a right triangle) and derive the result. In CBSE exams, starting from the result and "proving" it in a circle gets zero marks.