Question

The expansion $\frac{\sin \frac{7\pi }{24}+\sin \frac{5\pi }{24}+\sin \frac{9\pi }{24}+\sin \frac{3\pi }{24}}{\cos \frac{7\pi }{24}+\cos \frac{5\pi }{24}+\cos \frac{9\pi }{24}+\cos \frac{3\pi }{24}}$

• A. $1$
• B. $2-\sqrt{3}$
• C. $\sqrt{2}-1$
• D. $\frac{1}{\sqrt{3}}$

Answer 1

The correct option is A $1$

$sin\frac{7\pi }{24}+sin\left(\frac{5\pi }{24}\right)+sin\frac{9\pi }{24}+sin\frac{3\pi }{24}\Rightarrow sin(\frac{\pi }{2}-\frac{5\pi }{24})+sin(\frac{\pi }{2}-\frac{7\pi }{24})+sin(\frac{\pi }{2}-\frac{9\pi }{24})+sin(\frac{\pi }{2}-\frac{3\pi }{24})$

$\Rightarrow cos\frac{7\pi }{24}+cos\frac{5\pi }{24}+cos\frac{3\pi }{24}+cos\frac{9\pi }{24}.$

$\therefore \frac{sin\frac{7\pi }{24}+sin\left(\frac{5\pi }{24}\right)+sin\frac{9\pi }{24}+sin\frac{3\pi }{24}}{cos\frac{7\pi }{24}+cos\left(\frac{5\pi }{24}\right)+sin\frac{9\pi }{24}+sin\frac{3\pi }{24}}=1$