Question

If $\cos \theta =\frac{1}{2}\left(a+\frac{1}{a}\right)$, then $\cos 3\theta$ in terms of ' $a$ ' is

• A. $\frac{1}{4}\left(a^3+\frac{1}{a^3}\right)$
• B. $\frac{1}{2}\left(a^3+\frac{1}{a^3}\right)$
• C. $4\left(a^3+\frac{1}{a^3}\right)$
• D. $\left(a^3+\frac{1}{a^3}\right)$

Answer 1

Answer: (B)

$\cos 3\theta =4{\cos }^3\theta -3\cos \theta =\cos \theta \left(4{\cos }^2\theta -3\right)$
$=\frac{1}{2}\left(a+\frac{1}{a}\right)\left\{4\times \frac{1}{4}{\left(a+\frac{1}{a}\right)}^2-3\right\}=\frac{1}{2}\left(a+\frac{1}{a}\right)\left\{a^2+\frac{1}{a^2}-1\right\}=\frac{1}{2}\left(a^3+\frac{1}{a^3}\right)$