Question

The number of distinct solutions of sin5$\theta$.cos3$\theta$ = sin9$\theta$.cos7$\theta$ in [0, $\frac{\pi }{2}$] is

• A. 4
• B. 5
• C. 8
• D. 9

Answer 1

The correct option is D

9

sin8$\theta$ + sin2$\theta$ = sin16$\theta$ + sin2$\theta$

sin16$\theta$ = sin8$\theta$

∴ 16$\theta$ = n$\pi$ + $(-1{)}^n$8$\theta$

⇒ 8$\theta$ = 2m$\pi$ when n is even

⇒ 24$\theta$ = (2m + 1)$\pi$ when n is odd

∴ $\theta$ = $\frac{m\pi }{4}$, $\frac{(2m+1)\pi }{24}$, when m ∈ I

= 0, $\frac{\pi }{2}$, $\frac{\pi }{4}$ and $\frac{\pi }{24}$, $\frac{\pi }{8}$, $\frac{5\pi }{24}$, $\frac{9\pi }{24}$, $\frac{7\pi }{24}$, $\frac{11\pi }{24}$