Answer
Yes, No; see explanations.
Work Step by Step
Step 1. Given the identity
$sin(x+a)=sin(x)cos(a)+cos(x)sin(a)$,
we can differentiate both sides to get:
LHS=$cos(x+a)$
and
RHS=$cos(x)cos(a)-sin(x)sin(a)=cos(x+a)$
Step 2. Thus, LHS=RHS for all $x$ and the derivative is also an identity.
Step 3. Given an equation $x^2-2x-8=0$, we try to take derivatives of both sides:
LHS=$2x-2$ and RHS=$0$
Step 4. As the LHS=RHS is true for only $x=1$, the result is not an identity; the principle does not apply to this case.
