TrigExpand Command

TrigExpand( <Expression> ): Expands trigonometric functions of a sum of variables into trigonometric functions of a single variable, or expands products of trigonometric functions into linear expressions.

TrigExpand(tan(x + y)) gives \(\frac{\frac{\sin(x)}{\cos(x)}+\frac{\sin(y)}{\cos(y)}} {1-\frac{\sin(x)}{\cos(x)} \cdot \frac{\sin(y)}{\cos(y)}} \).

TrigExpand(sin(x)sin(x/3)) gives \(\frac{1}{2} \; \cos \left( 2 \cdot \frac{x}{3} \right) - \frac{1}{2} \; \cos \left( 4 \cdot \frac{x}{3} \right) \).

TrigExpand( <Expression>, <Target Function> ): Expands trigonometric functions of a sum of variables into expressions containing (when possible) the given target function of a single variable.

TrigExpand(tan(x + y), tan(x)) gives \(\frac{-\tan(x) - \tan(y)}{\tan(x) \tan(y) - 1}\).

CAS Syntax

CAS syntaxes may show different results, depending on the selected output mode:

TrigExpand(tan(x + y)) in Evaluate mode gives \(\frac{\frac{\sin(x)}{\cos(x)}+\frac{\sin(y)}{\cos(y)}} {1-\frac{\sin(x)}{\cos(x)} \cdot \frac{\sin(y)}{\cos(y)}} \) in Numeric mode gives \(\frac{\sin(x) \cos(y) + \sin(y) \cos(x)}{- \sin(x) \sin(y) +\cos(x) \cos(y) }\) .

The following commands are only available in the CAS View:

TrigExpand( <Expression>, <Target Function>, <Target Variable> ): Expands trigonometric expressions into expressions containing (when possible) only the given target function and variable.

TrigExpand(sin(x), sin(x), x/2) gives \(2\cos \left( \frac{x}{2} \right) \sin \left( \frac{x}{2} \right) \)
TrigExpand(sin(x)/(1+cos(x)), tan(x), x/2) gives \(\tan \left( \frac{x}{2} \right)\).

TrigExpand( <Expression>, <Target Function>, <Target Variable>, <Target Variable> ): Expands trigonometric expressions into expressions containing (when possible) only the given target function and variables.

TrigExpand(csc(x) - cot(x) + csc(y) - cot(y), tan(x), x/2, y/2) gives \(\tan \left( \frac{x}{2} \right) +\tan \left( \frac{y}{2} \right) \).