ɑ and ẞ are two positive acute angles and cos 2ɑ= 3cos⁡2β−13−cos⁡2β \frac{3 \cos{2} β - 1}{3 - \cos{2} β} Which of the following relation is correct?

Question

ɑ and ẞ are two positive acute angles and cos 2ɑ=  3cos⁡2β−13−cos⁡2β \frac{3 \cos{2} β - 1}{3 - \cos{2} β}  Which of the following relation is correct?

Accepted Answer

ঘ. tanɑ= √2 tanẞ

ব্যাখ্যা: দেওয়া আছে, $\cos 2\alpha = \frac{3 \cos 2\beta - 1}{3 - \cos 2\beta}$।
বিয়োজন-যোজন (componendo and dividendo) করে পাই:
$$\frac{1 - \cos 2\alpha}{1 + \cos 2\alpha} = \frac{(3 - \cos 2\beta) - (3 \cos 2\beta - 1)}{(3 - \cos 2\beta) + (3 \cos 2\beta - 1)}$$
$$\implies \frac{2 \sin^2 \alpha}{2 \cos^2 \alpha} = \frac{4 - 4 \cos 2\beta}{2 + 2 \cos 2\beta} = \frac{4(1 - \cos 2\beta)}{2(1 + \cos 2\beta)}$$
$$\implies 	an^2 \alpha = 2 	an^2 \beta$$
যেহেতু $\alpha$ ও $\beta$ ধনাত্মক সূক্ষ্মকোণ, তাই উভয়পাশে বর্গমূল নিলে পাই:
$$	an \alpha = \sqrt{2} 	an \beta$$

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