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cos(s + t) and 1 + tan(x)^2 = sec(x)^2 are not proved, though their sine and cotangent twins are #2142
Description
Simplify proves one identity of each pair below and not the other:
| Proved | Not proved |
|---|---|
sin(s + t) - sin(s)*cos(t) - cos(s)*sin(t) → 0 |
cos(s + t) - cos(s)*cos(t) + sin(s)*sin(t) unchanged |
1 + cot(x)^2 - cosec(x)^2 → 0 provided not sin(x) = 0 |
1 + tan(x)^2 - sec(x)^2 unchanged |
A reader of the reference chapter of Modenov's Problems in Geometry (#2131) also found that the tangent addition formula and the half-angle, Weierstrass, power-reduction, sum-to-product and product-to-sum formulas fail. That makes 13 of the chapter's trigonometric identities.
Related: #2135 (shifts by a multiple of π/2).
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Rafael-SOWNet commentedon Oct 27, 2026
The cosine half is fixed by #2161:
cos(s + t)andcos(s - t)now expand as the sine already did, socos(s + t) - cos(s)*cos(t) + sin(s)*sin(t)is0.1 + tan(x)^2 = sec(x)^2and the tangent, half-angle, power-reduction, sum-to-product and product-to-sum identities are still open here.One and a squared tangent are found anywhere in a sum (#2176)
Rafael-SOWNet commentedon Oct 27, 2026
1 + tan(x)^2 - sec(x)^2is0 provided not cos(x) = 0since #2176. The identity and its cotangent twin are now also found wherever the1and the square sit in a sum, so the sort order no longer decides. Still open here: the tangent addition formula, half-angle, Weierstrass, sum-to-product and product-to-sum identities, and the cosine power reductioncos(x)^2 - (1 + cos(2x))/2. Its sine twin is answered, and its candidates reach-1/2 + cos(x)^2/2 + sin(x)^2/2, where the squares are quotients that the Pythagorean rules do not match.The power reduction of a squared cosine is proved, as the squared sin…