Pythagorean identities — sin² + cos² = 1, derivations of the other two

3.1.9 · Maths › Advanced Trigonometry

HUM KYA CLAIM kar rahe hain?

"Pythagorean" word literal hai: har ek identity theorem $a^2+b^2=c^2$ hi hai, bas disguise mein.

PEHLI identity TRUE kyun hai? (Scratch se derivation)

KAISE milti hai identity 1:

1. Circle par point: $P=(\cos \theta ,\sin \theta )$. Ye step kyun? Unit-circle coordinates ki definition se.
2. $x$-axis par perpendicular drop karo. Hume ek right triangle milta hai jisme:
   • horizontal leg $=|\cos \theta |$
   • vertical leg $=|\sin \theta |$
   • hypotenuse $=$ radius $=1$. Ye step kyun? Radius origin se $P$ tak ki straight-line distance hai, jo hypotenuse hai.
3. Pythagoras lagao: $({\text{leg}}_1{)}^2+({\text{leg}}_2{)}^2=(\text{hyp}{)}^2$: ${\cos }^2\theta +{\sin }^2\theta =1^2=1.$ Ye step kyun? Square karne se absolute values khatam ho jaati hain, toh ye har quadrant mein kaam karta hai, sirf pehle mein nahi.

Baaki dono derive karna — bas DIVIDE karo

Trick: ${\sin }^2\theta +{\cos }^2\theta =1$ lo aur poori equation ko kisi ek squared function se divide kar do.

Child 1: $1+{\tan }^2\theta ={\sec }^2\theta$

${\cos }^2\theta$ se divide karo (require karta hai $\cos \theta \ne 0$):

$\frac{{\sin }^2\theta }{{\cos }^2\theta }+\frac{{\cos }^2\theta }{{\cos }^2\theta }=\frac{1}{{\cos }^2\theta }$

Ye step kyun? Humne ${\cos }^2\theta$ isliye choose kiya kyunki $\frac{\sin \theta }{\cos \theta }=\tan \theta$ aur $\frac{1}{\cos \theta }=\sec \theta$ — exactly wahi functions jo chahiye the.

${\tan }^2\theta +1={\sec }^2\theta ⟹\boxed{1+{\tan }^2\theta ={\sec }^2\theta }$

Child 2: $1+{\cot }^2\theta ={\csc }^2\theta$

${\sin }^2\theta$ se divide karo (require karta hai $\sin \theta \ne 0$):

$\frac{{\sin }^2\theta }{{\sin }^2\theta }+\frac{{\cos }^2\theta }{{\sin }^2\theta }=\frac{1}{{\sin }^2\theta }$

Ye step kyun? ${\sin }^2\theta$ se divide karne par $\cot =\frac{\cos }{\sin }$ aur $\csc =\frac{1}{\sin }$ milta hai.

$1+{\cot }^2\theta ={\csc }^2\theta ⟹\boxed{1+{\cot }^2\theta ={\csc }^2\theta }$

Worked examples

Common mistakes

Forecast-then-Verify

Flashcards

Connections

• Unit Circle definition of sin and cos
• Pythagoras Theorem
• Reciprocal and Quotient trig identities ($\tan ,\sec ,\cot ,\csc$ define karne ke liye zaroori)
• Trig identity proofs — strategy
• Double angle formulas (${\cos }^2=1-{\sin }^2$ baar baar use hota hai)
• Solving trig equations

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