(a) Prove that $$ \cos (\alpha+\beta)+\cos (\alpha-\beta)=2 \cos \alpha \cos \beta $$.

Question

UpStudy AI · Accepted Answer

We start by recalling the cosine addition and subtraction formulas:

cos(α + β) = cos α cos β − sin α sin β  
  cos(α − β) = cos α cos β + sin α sin β

Now, add the two equations together:

cos(α + β) + cos(α − β)  
    = [cos α cos β − sin α sin β] + [cos α cos β + sin α sin β]

Notice that the terms involving sin α sin β cancel each other out, leaving:

= cos α cos β + cos α cos β  
  = 2 cos α cos β

Thus, we have proven that

cos(α + β) + cos(α − β) = 2 cos α cos β.
$$ \cos (\alpha+\beta)+\cos (\alpha-\beta)=2 \cos \alpha \cos \beta $$.

(a) Prove that \( \cos (\alpha+\beta)+\cos (\alpha-\beta)=2 \cos \alpha \cos \beta \).