যদি $\cos \theta+\sec \theta=2$ হয়, তবে $\cos ^{3} \theta+\sec ^{3} \theta$ এর মান কত?

$\cos \theta+\sec \theta=2$ $\Rightarrow \cos \theta+\frac{1}{\cos \theta}=2$ $\Rightarrow \cos ^{2}$ $\theta+1=2 \cos \theta$ $\Rightarrow \cos ^{2}$ $\theta-2 \cos \theta+1=0$ $\Rightarrow(\cos \theta-1)^{2}=0$ $\therefore \cos \theta=1$ $\therefore \sec \theta=1$ $\therefore \cos ^{3} \theta+\sec ^{3}$ $\theta=1^{3}+1^{3}$ $=1+1=2$ $\text { Shortcut: }$ $\cos ^{3} \theta+\sec ^{3} \theta$ $=(\cos \theta+\sec \theta)^{3}-3$ $\cdot \cos \theta \sec \theta(\cos \theta+\sec \theta)$ $=2^{3}-3 \cdot 1 \cdot 2$ $=8-6=2$