Question

sin²A+sin²(A+2π÷3)+sin²(A+4π÷3)=3÷2​

Answer 1

✪ᴀɴsᴡᴇʀ✪

❥︎❥︎ $sin^2(A) + sin^2(A + \frac{2\pi}{3} )$

$\:\:\:\:\:+ sin^2(A + \frac{4\pi}{3} )$

❥︎❥︎ $sin^2(A)+ sin^2(\pi +A - \frac{\pi}{3} )$

$\:\:\:\:\:+ sin^2(\pi + A + \frac{\pi}{3} )$

❥︎❥︎ $sin^2(A) + sin^2(A - \frac{\pi}{3} )$

$\:\:\:\:\:+ sin^2(A + \frac{\pi}{3} )$

❥︎❥︎ $sin^2(A)$

$+[sin(A)cos(\frac{2\pi}{3} ) - cos(A)sin(\frac{2\pi}{3})]^{2}$

$+ [sin(A)cos(\frac{2\pi}{3} ) + cos(A)sin(\frac{2\pi}{3})]^{2}$

❥︎❥︎ $sin^2(A)+ 2[sin^{2} (A)cos^{2} (\frac{2\pi}{3})$

$\:\:\:\:\:+ cos^{2} (A)sin^{2} (\frac{2\pi}{3})]$

❥︎❥︎ $sin^2(A)+2[sin^{2} (A) \frac{1}{4}$

$\:\:\:\:\:+ cos^{2}(A) \frac{3}{4}]$

❥︎❥︎ $sin^2(A)+sin^{2} (A) \frac{1}{2} + cos^{2}(A) \frac{3}{2}$

❥︎❥︎ $\frac{3}{2}sin^2(A) + \frac{3}{2} cos^{2}(A)$

❥︎❥︎ $\frac{3}{2}[sin^2(A) + cos^{2}(A)]$

❥︎❥︎ $\frac{3}{2}$

Hence Proved