Question

The solution of the trigonometric equation ${\cos }^2\left(\frac{\pi }{3}\cos x-\frac{8\pi }{3}\right)=1$ must be

• A. ${\cos }^{-1}(3n+8)$
• B. ${\cos }^{-1}(3n-8)$
• C. $2n\pi$
• D. None of these

Answer 1

Answer: (C)

The equation is same as $2{\cos }^2(t)=2$
where $t=\frac{\pi }{3}\cos x-\frac{8\pi }{3}$
$\Rightarrow 1+\cos 2t=2$
$\Rightarrow \cos 2t=1$
$\Rightarrow 2t=2k\pi$
$\Rightarrow t=k\pi$
$\Rightarrow \frac{\pi }{3}\cos x-\frac{8\pi }{3}=k\pi$
$\Rightarrow \cos x=3k+8$
Since $|\cos x|\le 1$, we must have $|3k+8|\le 1$
$\Rightarrow k=-1$
$\Rightarrow \cos x=1$
$\Rightarrow x=2k\pi$
Thus choices (a), (b) are ruled out and (c) is correct.