1. State the following product-to-sum formulas:
    sin⁡(s) cos⁡(t)=\displaystyle {\sin{{\left({s}\right)}}}\ {\cos{{\left({t}\right)}}}=  
    cos⁡(s) cos⁡(t)=\displaystyle {\cos{{\left({s}\right)}}}\ {\cos{{\left({t}\right)}}}=  
    sin⁡(s) sin⁡(t)=\displaystyle {\sin{{\left({s}\right)}}}\ {\sin{{\left({t}\right)}}}=  

    Remark: The formulas allow us to write the periodic functions as a sum of simple harmonic functions. They are quite useful for handling calculus problems, and the uniqueness of such expressions is critical for decomposing signals in physics.

  2. Prove the three product-to-sum formulas.
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