I don't know how to prove sin^2(x)+cos^2(x)=1

I want to prove this thing by Matlab

sin^2 (x) + cos^2 (x) = 1

Somebody help me.

Please.....

[Bruno Luong]

Nothing can be prove by MATLAB or any software

Answer 1

[KSSV]

There are multiple ways to prove this:

Option 1:

x = rand ;
sin(x)^2+cos(x)^2 

Option 2:

syms a b c 

sinx = a/c ;
cosx = b/c ;
LHS = sinx^2+cosx^2 ;

In the above we have

Thus LHS = 1

Answer 2

[Ameer Hamza]

If you have symbolic toolbox, you can use isAlways()

syms x

eq = sin(x)^2 + cos(x)^2 == 1;
tf = isAlways(eq)

value of 1 means that it is correct for all values of 'x'.

Answer 3

[James Tursa]

[cos(x),sin(x)] is defined to be a point on the unit circle, so by definition we have sin^2(x) + cos^2(x) = 1 always. This isn't something to be proved since it is a definition. If you want to demonstrate it with values, you can always just plug stuff in and see that you always get about 1 within numerical floating point errors, or make x symbolic and evaluate the expression. But, this will just be a "feel good" demonstration ... it doesn't prove anything since the result follows directly from the definition.

Answer 4

[Bruno Luong]

Definition

exp(x) = sum x^k/factorial(k).

Property (provable here)

exp(x+y) = exp(x)*exp(y)

Definitions

sin(x) = 1/(2i)  (exp(ix) - exp(-ix))
cos(x) = 1/2 (exp(ix) + exp(-ix))

Thus using exp(0) = 1, i^2=-1

sin(x)^2 = -1/4 (exp(2ix) + exp(-2ix) -2)
cos(x)^2 = +1/4 (exp(2ix) + exp(-2ix) +2)

When sum it all the exp terms go away and we get

sin(x)^2 + cos(x)^2 = 1

(then one can deduce (sin(x),cos(x)) is point on circle, NOT the opposite definition of sin(x) is abcissa coordinate of point on circle bla bla bla).