Монгол Бодлогын Сан

Заавар: Хүртвэрийг $\cos^2\alpha+\sin^2\alpha=1$ тоогоор үржиж зэргийг өсгө.

Бодолт: $$\begin{align*} \dfrac{\cos\alpha+3\sin\alpha}{\cos^3\alpha-\sin^3\alpha}&=\dfrac{(\cos\alpha+3\sin\alpha)(\cos^2\alpha+\sin^2\alpha)}{\cos^3\alpha-\sin^3\alpha}\\ &=\dfrac{\cos^3\alpha+\cos\alpha\cdot\sin^2\alpha+3\sin\alpha\cos^2\alpha+3\sin^2\alpha}{\cos^3\alpha-\sin^3\alpha} & \color{red}{\dfrac{:\sin^3\alpha}{:\sin^3\alpha}}\\ &=\dfrac{\ctg^3\alpha+\ctg\alpha+3\ctg^2\alpha+3}{\ctg^3\alpha-1}\\ &=\dfrac{2^3+2+3\cdot 2^2+3}{2^3-1}=\dfrac{25}{7} \end{align*}$$