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  4. If x + (1/x) = 2 cos θ , then x3 + (1/x3) is :

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Q. If $x + \frac{1}{x} = 2 \cos \theta $, then $x^3 + \frac{1}{x^3}$ is :

Trigonometric Functions

Solution:

Given : $x + \frac{1}{x} = 2 \cos \theta$ ....(1)
Cubic both sides in $eq^n$ (1) we get
$x^{3} + \frac{1}{x^{3}} + 3 \left(x+ \frac{1}{x}\right) = 8 \cos^{3} \theta$
$ \Rightarrow x^{3} + \frac{1}{x^{3} } + 3\left(2\cos\theta\right) = 8 \cos^{3} \theta$
$ \Rightarrow x^{3} + \frac{1}{x^{3}} = 8 \cos^{3} \theta - 6 \cos\theta $
$\Rightarrow x^{3} + \frac{1}{x^{3} } = 2\left(4 \cos^{3} \theta - 3 \cos\theta\right) = 2 \cos3 \theta $