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sin(a+b) = sin(a)cos(b) + sin(b)cos(a) so
sin(2x+1) = sin(2x)cos(1) + sin(1)cos(2x) = cos(1)sin(2x) + sin(1)cos(2x) and
cos(1)sin(2x) + sin(1)cos(2x) `!=` sin(2x)+1. Just take the value x=0 then
cos(1)sin(0) + sin(1)cos(0) = sin(0) + 1 this simplifies to
sin(1) = 1 which is not true.
The other way is to take an individual value such as x=pi/4
sin(2(pi/4)+1) = sin(2pi/4) + 1
Simplify both sides to get
sin(pi/2 + 1) = sin(pi/2) + 1 sin(pi/2) = 1 so we have
sin(pi/2 + 1) = 1 + 1 = 2
And we know that sin(a) <= 1 for all real a, so no matter what sin(pi/2 + 1) is it cannot be equal to 2. This is a counterexample and proves that
sin(2x+1) `!=` sin(2x) + sin(1) is not an identity.
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