Question

Sin^2x + cos^2x = 1 proof in short

Answer 1

Take a right angle triangle ABC with ∠B = 90°.   So AC is the hypotenuse.
  So  Pythagoras theorem says:  AB² + BC² = AC²     --- (1)

Sin A = BC /AC              Cos A = AB/AC       by definition.

Sin² A + cos² A = BC²/AC²  +    AB² / AC² =  1          using (1).

proved.

Answer 2

Use Pythagorean Theorem:

We know,

In the right triangle,

Cosθ = $\frac{h}{r}$
Sinθ = $\frac{v}{r}$

where h means horizontal
           v means vertical 
           r means hypotenuse

∴ Pythagorean Theorem,

r² = v²+h² = r²sinθ + r²cos²θ ⇔ cos²θ+sin²θ = 1