The technique when "solving" a trig identity is to start with the identity and work on both sides until you get the two sides equal. The "proof" of the identity then proceeds by starting with this final, true statement and progressing backwards, through the sequence of steps you have, to arrive at the identity. In the first part of this process you can of course work on both sides as long as each resulting equation is true if and only if the previous one is true. There are obvious pitfalls in this type of argument and the two column format is used as a preventative measure.
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