(a) Expand $\Rightarrow2+2\cos(\alpha-\beta)=2\cdot2\cos^2\!\big(\tfrac{\alpha-\beta}{2}\big)=4\cos^2\!\big(\tfrac{\alpha-\beta}{2}\big)$. (b) $Q^*=\sqrt{2DC_o/C_h}=2000$ units; the five inventory quantities follow.

Part (a) — Trigonometric identity

To prove: $(\cos\alpha+\cos\beta)^2+(\sin\alpha+\sin\beta)^2=4\cos^2\!\left(\dfrac{\alpha-\beta}{2}\right).$

Step 1 — expand both squares.

$\cos^2\alpha+2\cos\alpha\cos\beta+\cos^2\beta+\sin^2\alpha+2\sin\alpha\sin\beta+\sin^2\beta.$

Step 2 — group using $\cos^2\theta+\sin^2\theta=1$.

$=(\cos^2\alpha+\sin^2\alpha)+(\cos^2\beta+\sin^2\beta)+2(\cos\alpha\cos\beta+\sin\alpha\sin\beta).$

$=1+1+2\cos(\alpha-\beta)=2+2\cos(\alpha-\beta).$

Step 3 — apply $1+\cos\theta=2\cos^2(\theta/2)$.

$=2\big(1+\cos(\alpha-\beta)\big)=2\cdot2\cos^2\!\left(\frac{\alpha-\beta}{2}\right)=4\cos^2\!\left(\frac{\alpha-\beta}{2}\right).$

Part (b) — Economic Order Quantity (EOQ)

Given: annual demand $D=800$, unit price $=\₹0.02$, ordering cost $C_o=\₹5$/order, holding cost $C_h=10\%$ of unit price $=0.10\times0.02=\₹0.002$ per unit per year.

1. EOQ in units. $Q^*=\sqrt{\frac{2DC_o}{C_h}}=\sqrt{\frac{2(800)(5)}{0.002}}=\sqrt{\frac{8000}{0.002}}=\sqrt{4000000}=2000\text{ units}.$
2. Minimum inventory cost (ordering $+$ carrying): …