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### 10.3 Simplifying Formulas

The sections below describe all the various kinds of simplifications Calc provides in full detail. None of Calc’s simplification commands are designed to pull rabbits out of hats; they simply apply certain specific rules to put formulas into less redundant or more pleasing forms. Serious algebra in Calc must be done manually, usually with a combination of selections and rewrite rules. See Rearranging with Selections. See Rewrite Rules.

See Simplification Modes, for commands to control what level of simplification occurs automatically. Normally the algebraic simplifications described below occur. If you have turned on a simplification mode which does not do these algebraic simplifications, you can still apply them to a formula with the `a s` (`calc-simplify`) [`simplify`] command.

There are some simplifications that, while sometimes useful, are never done automatically. For example, the `I` prefix can be given to `a s`; the `I a s` command will change any trigonometric function to the appropriate combination of ‘`sin`’s and ‘`cos`’s before simplifying. This can be useful in simplifying even mildly complicated trigonometric expressions. For example, while the algebraic simplifications can reduce ‘`sin(x) csc(x)`’ to ‘`1`’, they will not simplify ‘`sin(x)^2 csc(x)`’. The command `I a s` can be used to simplify this latter expression; it will transform ‘`sin(x)^2 csc(x)`’ into ‘`sin(x)`’. However, `I a s` will also perform some “simplifications” which may not be desired; for example, it will transform ‘`tan(x)^2`’ into ‘`sin(x)^2 / cos(x)^2`’. The Hyperbolic prefix `H` can be used similarly; the `H a s` will replace any hyperbolic functions in the formula with the appropriate combinations of ‘`sinh`’s and ‘`cosh`’s before simplifying.

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