Wave equation for a standing wave - clarification needed

Attachment 58158

Both of the highlighted equations deal with a standing wave. However, they are slightly different in the sense that the latter has a phase shift in it.

Why ?

Also, how does one go from the latter equation for a standing wave to :

2A*sinkx*sinwt. For a string with both ends fixed

And

2A*sinkx*coswt. For one end fixed and the other free

And

2A*coskx*coswt. For both free.

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