We want to see where Oliver is making a mistake in his claim.
The correct option is the first one, the simplification on the last step is wrong.
We start with the expression:
[tex]tan( \frac{3*pi}{4} + x)[/tex]
Now, by looking at Olvier's work, we can see that on the last step he writes:
[tex]\frac{-1 + tan(x)}{1 + tan(x)} = -1[/tex]
This is clearly wrong, as the numerator and the denominator are dont simplify to -1 (this only happens if the numerator is -1 times the denominator), so the left side can't be equal to -1, it actually would be simplified to:
[tex]\frac{-1 + tan(x)}{1 + tan(x)} = \frac{-1 +1 - 1 tan(x)}{1 + tan(x)} = \frac{-2 + 1 + tan(x)}{1 + tan(x)} \\\\\frac{-2 + (1 + tan(x))}{1 + tan(x)} = \frac{-2 }{1 + tan(x)} + \frac{ 1 + tan(x)}{1 + tan(x)} = \frac{-2 }{1 + tan(x)} + 1[/tex]
Then the correct option is the first option.
If you want to learn more, you can read:
https://brainly.com/question/14324500
