Let $A, B, C$ be angles in a triangle. Is the following inequality $$4\cos A \le 1 + \cos\left(\frac{B-C}{2}\right)$$ true? I just assume it but don't have a proof. Thank you for your help.
2026-05-15 01:29:09.1778808549
Trigonometric inequality for angles in triangle
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Nope it is not true.
The right side is at most 2, while when $A$ is very small the left side is close to 4.
Note that if $0 <A< 30^\circ$ then
$$4 \cos(A) > 2 \geq 1 + \cos\left(\frac{B-C}{2}\right)$$