How to prove that sin^2 x+cos ^2 x=1 using derivatives?
We'll assign a function to the Pythagorean expression (sin x)^2 + (cos x)^2 = f(x).
From Pythagorean identity, we notice that f(x) = 1.
We also know that the derivative of a constant function is cancelling.
Therefore, if we'll differentiate f(x) with respect to x, the result has to be zero.
Let's check if it's true.
f'(x) = 2 sin x*(sin x)' + 2*cos x*(cos x)'
f'(x) = 2sin x*cos x - 2 cos x*sin x
We'll eliminate like terms and we'll get:
f'(x) = 0
This result emphasize the fact that the function is a constant.
Let's see if the sum of the squares of the sine and cosine functions of the same angle is 1.
We'll put x = 0.
(sin 0)^2 + (cos 0)^2 = 0 + 1 = 1
We'll put x = pi/2
(sin pi/2)^2 + (cos pi/2)^2 = 1 + 0 = 1
We'll put x = pi
(sin pi)^2 + (cos pi)^2 = 0 + (-1)^2 = 1
We'll put x = 2pi
(sin 2pi)^2 + (cos 2pi)^2 = 0 + 1 = 1
We notice that the Pythagorean identity, (sin x)^2 + (cos x)^2 = 1, is verified for any value of x.