3. trignometrical ratios,functions and identities — ગણિત ધોરણ 11 સાયન્સ — Question

Now ${x^3} + \frac{1}{{{x^3}}} = {\left( {x + \frac{1}{x}} \right)^3} - 3x\frac{1}{x}\left( {x + \frac{1}{x}} \right)$

$=  {(2\cos \theta )^3} - 3(2\cos \theta ) = 8{\cos ^3}\theta - 6\cos \theta $

$=  2(4{\cos ^3}\theta - 3\cos \theta ) = 2\cos 3\theta $.

Trick : Put $x = 1$ $ \Rightarrow $ $\theta = {0^\circ }$.

Then ${x^3} + \frac{1}{{{x^3}}} = 2 = 2\cos 3\theta $.