Question

prove that (sin 2theta + cos 2 theta=1)​

Answer 1

We have to prove that  sin² θ + cos² θ = 1.

This formula can easily be proven by a right triangle!

Consider a right triangle having an acute angle θ, where the third angle will be  90° - θ.

We get,

sin θ  =  (opposite side of θ) / (hypotenuse)

sin² θ  =  [(opposite side of θ) / (hypotenuse)]²  =  (opposite side of θ)² / (hypotenuse)²

And,

cos θ  =  (adjacent side of θ) / (hypotenuse)

cos² θ  =  [(adjacent side of θ) / (hypotenuse)]²  =  (adjacent side of θ)² / (hypotenuse)²

And we remember the famous Pythagoras' Theorem.

(opposite side of θ)² + (adjacent side of θ)² = (hypotenuse)²

Now,

     LHS

⇒  sin² θ + cos² θ

⇒  [(opposite side of θ)² / (hypotenuse)²]  +  [(adjacent side of θ)² / (hypotenuse)²]

⇒  [(opposite side of θ)² + (adjacent side of θ)²]  /  (hypotenuse)²

⇒  (hypotenuse)² / (hypotenuse)²

⇒  1

⇒  RHS

Hence Proved!