1. Prove the equation

sin2x+cos2x=1

2. Re: Prove the equation

Have you attempted a web search on the topic?

Prove the Pythagorean identity

3. Re: Prove the equation

Just draw a right triangle. Select and angle
fine sine and cosine of the selected angle in terms of sides.
Substitute the values in Left hand side of the equation and simplify
Keep in mind the Pythagoras Theorem

4. Re: Prove the equation

$\sin\,x=\dfrac{opposite}{hypotenuse}$

$\sin^2\,x= \left( \dfrac{opposite}{hypotenuse} \right)^2$

$\cos\,x=\dfrac{adjacent}{hypotenuse}$

$\cos^2\,x= \left (\dfrac{adjacent}{hypotenuse} \right)^2$

$\sin\,x+\cos^2\,x= \left (\dfrac{opposite}{hypotenuse} \right)^2+ \left (\dfrac{adjacent}{hypotenuse} \right)^2$

$= \left (\dfrac{(opposite)^2+(adjacent )^2}{(hypotenuse)^2} \right)$

$=\dfrac{(hypotenuse)^2}{(hypotenuse)^2}$

$=1$

5. Re: Prove the equation

There exist a number of different ways to prove that. How you prove it depends strongly on how you are defining sine and cosine- which of the many definitions of sine and cosine are you using?