Why does y=(sinx)^2+(cosx)^2 create a flat line on a graph?

Hello!

A flat line (a straight horizontal line) as a graph of `y=f(x)` means that `f(x)` is a constant (all its values are the same). Is this true for `y=sin^2(x)+cos^2(x)`? Yes, it is one of the most known trigonometric identities.

It is simple to prove it, at least for `x` between `0` and `pi/2` (90 degrees). Such an `x` may be an angle in a right triangle. If the lengths of its legs are `a` and `b,` then the length of the hypotenuse is `sqrt(a^2+b^2)` by the Pythagorean theorem.

Let `a` be the adjacent leg of the angle `x,` then

 `cos(x)=a/sqrt(a^2+b^2)`  and  `sin(x)=b/sqrt(a^2+b^2).`

Therefore  `cos^2(x)=a^2/(a^2+b^2)`  and  `sin^2(x)=b^2/(a^2+b^2),`

and  `cos^2(x)+sin^(x)=(a^2+b^2)/(a^2+b^2)=1.`

The answer is: because the given function is a constant (and this constant is 1).