Discussion
The absolute maximum value of \(z = \cos(x + y) + \cos(x - y)\) where \(x + y = 6\) is:
Answer: B
Use the sum to product identity, or simply note that the first term is fixed because \(x + y = 6\).
\[ z = \cos 6 + \cos(x - y) \]
The only term that varies is \(\cos(x - y)\), whose largest value is \(1\) (reached when \(x - y = 0\), that is \(x = y = 3\)). So the maximum is \(\cos 6 + 1 = 1 + \cos 6\). The option \(1 - \cos 6\) has the wrong sign on the fixed term, \(1 + \sin 2\) and \(1 + \cos 3\) do not equal \(1 + \cos 6\), since \(\cos 3 \neq \cos 6\). The answer is option B.
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