TrigFactor

The factoring counterpart of TrigExpand/TrigReduce — rewrites sums and differences of Sin/Cos into products.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageTrigFactor!↗

TrigFactor(expr)an identity for expr, factored via the sum-to-product/half-angle identities or, for a genuine polynomial in Sin(x0)/Cos(x0), via Factor after substitution.

Typed signatures

Type	Package	Applies
(expression) -> expression	analytic	always

Domain: Transformations

Details

• Two independent strategies, tried in order: (1) the four sum-to-product identities ($\sin a\pm\sin b$, $\cos a\pm\cos b$) and the two half-angle identities ($1\pm\cos x$), matched structurally (each confirmed against wolframscript's own TrigFactor, up to reordering); (2) failing that, if expr is a genuine polynomial in Sin(x0)/Cos(x0) for one common argument x0 (Add/Subtract/Negate/Multiply/an integer Power over those two atoms and x0-free numeric coefficients), substitute u = Sin(x0), v = Cos(x0), factor the polynomial in u, v with compute-engine's own Factor, and substitute back.
• The polynomial strategy is provably correct (whatever Factor returns for the polynomial IS one, so re-substituting reconstructs the identity) but does not chase the extra trig identities Wolfram's own TrigFactor sometimes prefers — e.g. it rewrites $\sin^2x-\cos^2x$ as $-2\sin(\pi/4-x)\sin(\pi/4+x)$ rather than the plain difference-of-squares $(\sin x-\cos x)(\sin x+\cos x)$ this file gives; both are correct identities, just not the same factorization.
• Declined: two DIFFERENT trig arguments mixed together outside the four named pairs (e.g. $\sin a\cos b$), any transcendental other than Sin/Cos, and an argument-dependent (non-integer) exponent.

Examples

Basic #

$\sin a+\sin b = 2\sin(\frac{a+b}2)\cos(\frac{a-b}2)$

live ↗

In$\mathrm{TrigFactor}(\sin(a)+\sin(b))$Out$2\sin(\frac{a+b}{2})\cos(\frac{a-b}{2})$

$\cos a-\cos b = -2\sin(\frac{a+b}2)\sin(\frac{a-b}2)$

live ↗

In$\mathrm{TrigFactor}(\cos(a)-\cos(b))$Out$-2\sin(\frac{a+b}{2})\sin(\frac{a-b}{2})$

$1-\cos x = 2\sin^2(x/2)$

live ↗

In$\mathrm{TrigFactor}(1-\cos(x))$Out$2\sin(\frac{x}{2})^2$

Scope #

the polynomial strategy — $\sin^2x+2\sin x\cos x+\cos^2x = (\sin x+\cos x)^2$

live ↗

In$\mathrm{TrigFactor}(\sin(x)^2+2\sin(x)\cos(x)+\cos(x)^2)$Out$(\sin(x)+\cos(x))^2$

A difference of squares factors as $(\sin x-\cos x)(\sin x+\cos x)$

live ↗

In$\mathrm{TrigFactor}(\sin(x)^2-\cos(x)^2)$Out$(\sin(x)-\cos(x))(\sin(x)+\cos(x))$

The factoring is applied inside, but the whole stays symbolic

live ↗

In$\mathrm{TrigFactor}(\frac{1}{\sinh(x)}+\frac{\cosh(x+y)}{\sin(x)^2-\cos(x)^2})$Out$\mathrm{TrigFactor}(\frac{1}{\sinh(x)}+\frac{\cosh(x+y)}{\sin(x)^2-\cos(x)^2})$

About

Examples	6
Checked against	Wolfram Language 5 agree1 differ
Crosswalk	Wolfram Language

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/trig-factor.ts

See also: ExpToTrig, FunctionExpand