Detail two methods that you could use to show that sin(2x + 1) = sin(2x) + sin 1 is not an identity. College Trig question

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.