Question

if √3tanA=3sinA, prove that (sin2 A-cos2 A) =1/3​

Answer 1

$= > \sqrt{3} \: tan \: a = 3 \: sin \: a \\ \\ = > \sqrt{3} \: \frac{sin \: a}{cos \: a} = 3 \: sin \: a \\ \\ = > \frac{1}{cos \: a} = \sqrt{3} \\ \\ = > cos \: a \: = \frac{1}{ \sqrt{3} } \\ \\ squaring \: both \: sides \\ \\ = > {cos}^{2} a \: = \frac{1}{3} \\ \\ = > 1 - {sin}^{2} a \: = \frac{1}{3} \\ \\ = > {sin}^{2} \: a = 1 - \frac{1}{3} \\ \\ = > {sin}^{2} \: a \: = \frac{2}{3} \\ \\ now \: \\ = {sin}^{2}a - {cos}^{2} a \\ \\ = \frac{2}{3} - \frac{1}{3} \\ \\ = \frac{2 - 1}{3} \\ \\ = \frac{1}{3} \\ \\ hence \: proved$