Question

Sin^2A + sin^2(A+pi/3) + sin^2(A-pi/3) = 3/2

Answer 1

hope it helps and you will understand

Answer 2

$i ) sin (A + \frac{\pi}{3} )\\= sinA cos \frac{\pi}{3} + sin \frac{\pi}{3} cosA\\= sin A \times \frac{1}{2} + \frac{\sqrt{3}}{2} cos A \\= \frac{1}{2} ( sinA + \sqrt{3} cosA) \: ---(1)$

$ii ) sin (A - \frac{\pi}{3} )\\= sinA cos \frac{\pi}{3} - sin \frac{\pi}{3} cosA\\= sin A \times \frac{1}{2} - \frac{\sqrt{3}}{2} cos A \\= \frac{1}{2} ( sinA - \sqrt{3} cosA) \: ---(2)$

$Now , LHS = \red{sin^{2} A + sin^{2} (A + \frac{\pi}{3} )+sin^{2}(A - \frac{\pi}{3} )}\\= sin^{2} A +[ \frac{1}{2} ( sinA + \sqrt{3} cosA) ]^{2} + [ \frac{1}{2} ( sinA - \sqrt{3} cosA) ]^{2}$

$= sin^{2} A + \frac{1}{4} [ ( sinA + \sqrt{3} cosA)^{2} + ( sinA - \sqrt{3} cosA)^{2} ]\\= sin^{2} A + \frac{1}{4} [ 2( sin^{2} A + (\sqrt{3} cos A)^{2} ]\\</p><p>= sin^{2} A + \frac{1}{2} [ sin^{2} A + 3cos^{2}A ]\\</p><p>= sin^{2} A + \frac{1}{2} [ sin^{2} A + 3( 1 - sin^{2}A )] \\= sin^{2} A + \frac{1}{2} [ sin^{2} A + 3 - 3sin^{2}A ] \\</p><p>= sin^{2} A + \frac{1}{2} [ 3 - 2sin^{2}A \\</p><p>= sin^{2} A + \frac{ 3}{2} - \frac{2}{2}sin^{2}A \\</p><p>= sin^{2} A + \frac{ 3}{2} - sin^{2}A \\</p><p>\green {= \frac{ 3}{2}}</p><p>= RHS$

$Hence \:proved$

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