Question

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

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

Answer 1

Answer: (C)

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