Module symbolic_trig 

Trigonometric simplification (Gap analysis §3.3) — identity-driven rewrites over the CAS’s Sin/Cos/Tan nodes that plain simplify cannot do (it knows only constant folding and algebraic identities).

Implemented identities, each value-preserving for all real inputs:

• Pythagorean: k·sin²(u) + k·cos²(u) → k (any common scalar k, either order), and 1 − sin²(u) → cos²(u), 1 − cos²(u) → sin²(u).
• Parity: sin(−u) → −sin(u), cos(−u) → cos(u), tan(−u) → −tan(u).
• Quotient: sin(u)/cos(u) → tan(u).

Applied bottom-up to a bounded fixpoint and interleaved with the base simplify, so the Pythagorean collapse also fires on nested sums after the algebra normalises them.

Functions

simplify_trig

Simplify trigonometric structure in expr, layered on the always-sound base simplify.