Question

If cos a/3 = 1/2 (p+1/2), prove that cos a=1/2(p^3+1/p^3)
$\cos \frac{ \alpha }{3} = \frac{1}{2} (p + \frac{1}{p} )$

$\cos( \alpha ) = \frac{1}{2} ({p}^{3} + \frac{1}{ {p}^{3} } )$
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Answer 1

Answer:

See the explanation

Step-by-step explanation:

Given

       $\cos\frac{\alpha}{3}=\frac{1}{2}(p+\frac{1}{p})$

Therefore using $\cos(3\theta)=4\cos^3\theta-3\cos\theta$

    $\cos\alpha=\cos3(\frac{\alpha}{3})=4\cos^3(\frac{\alpha}{3})-3\cos(\frac{\alpha}{3})$

                              $=4(\frac{1}{8}(p+\frac{1}{p})^3)-3(\frac{1}{2}(p+\frac{1}{p}))\\\\=\frac{1}{2}(p+\frac{1}{p})^3)-\frac{3}{2}(p+\frac{1}{p})\\\\=\frac{1}{2}(p^3+\frac{1}{p^3}+3p\times \frac{1}{p}(p+\frac{1}{p})-3(p+\frac{1}{p}))\\\\=\frac{1}{2}(p^3+\frac{1}{p^3}+3(p+\frac{1}{p})-3(p+\frac{1}{p}))\\=\frac{1}{2}(p^3+\frac{1}{p^3})$

Hence proved