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  4. If cos θ=(1/2)(a+(1/a)), then cos 3 θ in terms of ' a ' is

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

Trigonometric Functions

Solution:

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