Question

#QualityQuestion @Trigonometry \bf{Prove ~the~ identity~:}Prove the identity : (1 + cos(x) + cos(2x)) / (sin(x) + sin(2x)) = cot(x)​

Answer 1

Explanation:

$1 + \cos(x) + \cos(2x) = \cos {}^{2} ( x ) + \sin {}^{2} (x) + \cos(x) + cos {}^{2} ( x ) - \sin {}^{2} (x) \\ = 2 \cos {}^{2} (x) + \cos(x) = \cos(x) (2 \cos(x) + 1)$

$\sin(x) + \sin(2x) = \sin(x) + 2 \sin(x) \cos(x) \\ = \sin(x) (1 + 2 \cos(x) )$

$(1 + cosx + cos2x) \div (sinx + sin2x) \\ = (\cos(x) (1 + 2 \cos(x) )) \div ( \sin(x) (1 + 2 \cos(x) ) \\ = \cos(x) \div \sin(x) = \cot(x)$