Question:
If \(\cot\phi + \tan\phi = 4\) and \(\phi\) is an acute angle, find the value of \(\cot^3\phi + \tan^3\phi\).
यदि \(\cot\phi + \tan\phi = 4\) है और \(\phi\) एक न्यूनकोण है, तो \(\cot^3\phi + \tan^3\phi\) का मान ज्ञात कीजिए।
Options:
- 52
- 56
- 60
- 64
- 52
- 56
- 60
- 64
Correct Answer: a)
Explanation:
Let \(x = \cot\phi\), then \(\tan\phi = \dfrac{1}{x}\), so \(x + \dfrac{1}{x} = 4\).
\(x^3 + \dfrac{1}{x^3} = \left(x + \dfrac{1}{x}\right)^3 – 3\left(x + \dfrac{1}{x}\right) = 4^3 – 3(4) = 64 – 12 = 52\).
यदि \(x + \dfrac{1}{x} = 4\), तो \(x^3 + \dfrac{1}{x^3} = 4^3 – 3(4) = 64 – 12 = 52\)।