exponential form of the plane wave

Question

I am having a difficulty on understanding how we get from line one to line two:

$(1)$ $$1 + \exp(-ik( b_1 +b_2)) + \exp(-ikb_1) + \exp(-ikb_2)$$

$(2)$ $$\cos(k(b_1+b_2)/2) \cdot \Big(2\cos(k(b_1+b_2)/2) + 2\cos(k(b_1-b_2)/2)\Big)$$

Thanks.

Answer 1

Note that: $$ e^{ix} = \cos x + i \sin x $$ When looking at a wave equation and we want a mathematical description of the wave in the real world then we must look at the real part of the wave equation. $$ \Re{(1 + e^{-ik(b_1+b_2)} + e^{-ikb_1}+e^{-ikb_2})} = 1 + \cos k(b_1+b_2) + \cos kb_1 + \cos kb_2 = $$ $$ = \cos 0 + \cos k(b_1+b_2) + \cos kb_1 + \cos kb_2 $$ Using the sum to product formula for a cosine $$ \cos\alpha + \cos\beta = 2\cos\Big(\frac{\alpha+\beta}{2}\Big)\cos\Big(\frac{\alpha-\beta}{2}\Big) $$ for the first two terms and the second two terms we get the final result: $$ \cos\Big(\frac{k(b_1+b_2)}{2}\Big) \Big[2\cos\Big(\frac{k(b_1+b_2)}{2}\Big)+2\cos\Big(\frac{k(b_1-b_2)}{2}\Big)\Big]$$