Question

prove that cos 7π/12+cosπ/12=sin5π\12-sinπ/12​

Answer 1

Answer:

$\bf{cos\frac{7\pi}{12}+cos\frac{\pi}{12}=\bf{sin\frac{5\pi}{12}-sin\frac{\pi}{12}}$

Step-by-step explanation:

Prove that cos 7π/12+cosπ/12=sin5π\12-sinπ/12​

$\bf{cos\frac{7\pi}{12}+cos\frac{\pi}{12}}$

$=2\:cos(\frac{\frac{7\pi}{12}+\frac{\pi}{12}}{2})\:cos(\frac{\frac{7\pi}{12}-\frac{\pi}{12}}{2})$

$=2\:cos(\frac{\frac{8\pi}{12}}{2})\:cos(\frac{\frac{6\pi}{12}}{2})$

$=2\:cos(\frac{4\pi}{12})\:cos(\frac{3\pi}{12})$

$=2\:cos\frac{\pi}{3}\:cos\frac{\pi}{4}$

$=2(\frac{1}{2})(\frac{1}{\sqrt2})$

$=\frac{1}{\sqrt2}$..........(1)

$\bf{sin\frac{5\pi}{12}-sin\frac{\pi}{12}}$

$=2\:cos(\frac{\frac{5\pi}{12}+\frac{\pi}{12}}{2})\:sin(\frac{\frac{5\pi}{12}-\frac{\pi}{12}}{2})$

$=2\:cos(\frac{\frac{6\pi}{12}}{2})\:sin(\frac{\frac{4\pi}{12}}{2})$

$=2\:cos(\frac{3\pi}{12})\:sin(\frac{2\pi}{12})$

$=2\:cos\frac{\pi}{4}\:sin\frac{\pi}{6}$

$=2(\frac{1}{\sqrt2})(\frac{1}{2})$

$=\frac{1}{\sqrt2}$.........(2)

From (1) and (2)

$\bf{cos\frac{7\pi}{12}+cos\frac{\pi}{12}=\bf{sin\frac{5\pi}{12}-sin\frac{\pi}{12}}$

Answer 2

Answer:

Step-by-step explanation: