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Should we avoid using trig functions other than sine, cosine, and tangent? Should we think of facts about them (e.g., the other Pythagorean identities) as non-curricular facts? How about their definitions?
The progressions doc says “Because the second three may be expressed as reciprocals of the first three, this progression discusses only the first three,” but it’s unclear whether that means these are being left out of the discussion to save space, or that they’re being left out of the expectations for students because they’re redundant.
There is no prohibition on these in the standards, but they are not required. So the answer to your question really depends on entire curricular design. High school courses can include material not in the standards, but the material in the standards should come first. As a practical matter, I suspect for most students there is enough in the standards to take up a full year without including extra material, but an advanced course or a course for STEM-intending students might want to go further.
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