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    $\begingroup$ Note that you can get some nice looking wave-packets with this code: Plot[Evaluate[# - ReleaseHold[convert[#, Tan[x]]] &[ Sin[16 x] Cos[x]]], {x, 0, 4 Pi}] $\endgroup$ Commented Jan 21, 2012 at 12:47
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    $\begingroup$ @Nasser. Actually, I just looked at the maple link you supplied, and the functionality it gives is quite simple. I don't think it is even capable of the examples provided in the question (and my answer). convert(expr, tan) just uses the Weierstrass substitution step. convert(expr, sincos) just rewrites exp, tan, cot, etc as sin and cos. It does not simplify down to one function like the OP asked for. And so on with the other converts. $\endgroup$ Commented Jan 21, 2012 at 12:58
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    $\begingroup$ I find this really useful. Many thanks Simon! I wonder why it isn't built-in in Mathematica! $\endgroup$ Commented Jun 17, 2014 at 19:10
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    $\begingroup$ +1 for using "The world's sneakiest substitution" $\endgroup$ Commented May 27, 2015 at 15:35
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    $\begingroup$ @Histograms, it's really neat for algebra. A mixed bag in calculus, tho; bites the careless. $\endgroup$ Commented May 27, 2015 at 16:28