How can I verify the Pythagorean Trig Identity using approximations for $\sin x$ and $\cos x$ derived from their infinite series representations?
I can see that the infinite series of $\sin x$ and $\cos x$ should mean that for small angles, $\sin x$ is approximately $x$, and $\cos x$ is approximately $1-.5x^2$. I'd like to use these approximations to verify that $\sin^2x + \cos^2x$ is approximately $1$.
I seem to be off on the wrong track with my approach to this task. First, I simply substituted the approximations derived from the first (or first and second) terms of the series into the Pythagorean identity. Perhaps my algebra is off, but two attempts yielded the result $x^4 = 0$ which fails to confirm either the Pythagorean identity or the approximations. Of course, we know intuitively the results should verify truly.
Can you please suggest an appropriate way to approach this?
Added: I see from the answers below, that I should not have set the substitution in the Pythagorean Trig Identity equal to 1, because that very value of 1 is what I am verifying. Instead, the Pythagorean Trig Identity should be calculated using the approximation substitutions, then compared to the value of 1. If they are approximately close, the verification is done, and the Pythagorean Trig Identity is indeed approximately equal to 1.