Question

If $\cos 7 \theta=\cos \theta-\sin 4 \theta$, then the general value of $\theta$ is

• A. $\frac{n\pi}{6} ,\frac{n\pi}{3} + \left(-1\right)^{n} \frac{\pi}{18}$
• B. $\frac{n\pi}{3} ,\frac{n\pi}{3} + \left(-1\right)^{n} \frac{\pi}{18}$
• C. $\frac{n\pi}{4},\frac{n\pi}{3} \pm \frac{\pi}{18}$
• D. $\frac{n\pi}{4},\frac{n\pi}{3} + \left(-1\right)^{n} \frac{\pi}{18}$

Answer 1

Answer: (D)

$\cos 7 \theta=\cos \theta-\sin 4 \theta$
$p \sin 4 \theta=\cos \theta-\cos 7 \theta$
$p \sin 4 \theta=2 \sin 4 \theta \sin 3 \theta$
$p \sin 4 \theta(1-2 \sin 3 \theta)=0$
$\therefore \sin 4 \theta=0$ or $\sin 3 \theta=\frac{1}{2}$
$\Rightarrow 4 \theta=n \pi$ or $3 \theta=n \pi+(-1)^{n} \frac{\pi}{6}$
$\Rightarrow \theta=\frac{n \pi}{4}$ or $\frac{n \pi}{3}+(-1)^{n} \frac{\pi}{18}$