If you are starting from rational values and manipulating them, and want the results to be in the form you show, then the Symbolic Toolbox will generally do that.
But if you are starting from floating point numbers and want to find some representation in terms of rationals, sqrt() of rational, Pi, and exp() of rationals, then there is a lot of different ways to do that, most of which will be more or less accidental.
For example, sin(1) is about 0.841470984807897 which is not so different from sqrt(2) - pi/2 + 1 which is 0.843417235578199 ... more or less by chance.
sin(1) could also be expressed as being approximately
1 - 1/6 + 1/120 - 1/5040 + 1/362880 - 1/39916800 + 1/6227020800 - 1/1307674368000 + 1/355687428096000 - 1/121645100408832000 + 1/51090942171709440000 - 1/25852016738884976640000 + 1/15511210043330985984000000 - 1/10888869450418352160768000000 + 1/8841761993739701954543616000000 - 1/8222838654177922817725562880000000
which I derive from the taylor series.
but perhaps you would prefer
1 + 1/(-6 + 1/(-3 + 1/(-4 + 1/(-19))))
which is the output of rat(sin(1))...
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