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

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beckden | High School Teacher | (Level 1) Educator

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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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