Question

prove that cos^3acos3a+sin^3asina=cos^3 2a​

Answer 1

Step-by-step explanation:

We Have :-

$\cos^{3}(a) \: \cos(3a) + \sin^{3} (a) \sin(3a)$

To Prove :-

$\cos^{3}(a) \: \cos(3a) + \sin^{3} (a) \sin(3a) = \cos^{3} (2a)$

Solution :-

$\cos^{3}(a) \: \cos(3a) + \sin^{3} (a) \sin(3a) \\ \\ \\ we \: know \: that \: \\ \\ \\ \sin(3a) = 3 \sin(a) - 4 \sin^{3} (a) \\ \\ \\ \cos(3a) = 4 \cos^{3} (a) - 3 \cos(a) \\ \\ \\ using \: this \: in \: lhs \\ \\ \\ \cos^{3} (a) (4 \cos^{3} (a) - 3 \cos(a) ) + \sin^{3} (a) (3 \sin(a) - 4 \sin^{3} (a)) \\ \\ \\ 4 \cos^{6} (a) - 3 \cos^{4} (a) + 3 \sin^{4}(a) - 4 \sin^{6} (a) \\ \\ \\ 4( \cos^{6} (a) - \sin^{6} (a) ) - 3( \cos^{4} (a) - \sin^{4} (a) ) \\ \\ \\ 4( (\cos^{2} (a))^{3} - (\sin^{2} (a))^{3} ) - 3( (\cos^{2} (a) ) ^{2} - \sin^{2} (a))^{2} ) \\ \\ \\ 4( \cos^{2} (a) - \sin^{2} (a) )(( \cos^{2} (a))^{2} + ( \sin^{2} (a))^{2} + \cos^{2} (a) \sin^{2} (a) ) - 3( \cos^{2} (a) - \sin^{2} (a) )( \cos ^{2} (a) + \sin^{2} (a) ) \\ \\ \\ ( \cos^{2} (a) - \sin^{2} (a) )(4(( \cos^{2} (a))^{2} + ( \sin^{2} (a))^{2} + \cos^{2} (a) \sin^{2} (a) )) - 3\times 1 \\ \\ \\ ( \cos^{2} (a) - \sin^{2} (a) )((4\cos^{4} (a) + 4\sin^{4} (a) + 4 \cos^{2} (a) \sin^{2} (a) ) - 3( \cos^{2} (a) + \sin^{2} (a) ) ^{2} ) \\ \\ \\ ( \cos^{2} (a) - \sin^{2} (a) )((4\cos^{4} (a) + 4\sin^{4} (a) + 4 \cos^{2} (a) \sin^{2} (a) ) - 3( \cos^{4} (a) + \sin^{4} (a) + 2 \cos^{2} \sin^{2} (a) ) ) \\ \\ \\ ( \cos^{2} (a) - \sin^{2} (a) )((cos^{4} (a) + \sin^{4} (a) - 2 \cos^{2} \sin^{2} (a) ) ) \\ \\ \\ ( \cos^{2} (a) - \sin^{2} (a))( \cos^{2} (a) - \sin^{2} (a) ) ^{2} \\ \\ \\ ( \cos^{2} (a) - \sin^{2} (a) )^{3} \\ \\ \\( \cos(2a) )^{3} \\ \\ \\ \cos^{3} (2a) \\ \\ \\ = rhs \\ \\ \\ hence \: proved$

Answer 2

$\huge{\underline{\sf{Solution:-}}}$

$\cos^{3}(a) \: \cos(3a) + \sin^{3} (a) \sin(3a) \\ \\ We \: know \: that \: \\ \\ \sin(3a) = 3 \sin(a) - 4 \sin^{3} (a) \\ \\ \cos(3a) = 4 \cos^{3} (a) - 3 \cos(a) \\ \\ Using \: this \: in \: LHS \\ \\ \cos^{3} (a) (4 \cos^{3} (a) - 3 \cos(a) ) + \sin^{3} (a) (3 \sin(a) - 4 \sin^{3} (a)) \\ \\ 4 \cos^{6} (a) - 3 \cos^{4} (a) + 3 \sin^{4}(a) - 4 \sin^{6} (a) \\ \\ 4( \cos^{6} (a) - \sin^{6} (a) ) - 3( \cos^{4} (a) - \sin^{4} (a) ) \\ \\ 4( (\cos^{2} (a))^{3} - (\sin^{2} (a))^{3} ) - 3( (\cos^{2} (a) ) ^{2} - \sin^{2} (a))^{2} ) \\ \\ 4( \cos^{2} (a) - \sin^{2} (a) )(( \cos^{2} (a))^{2} + ( \sin^{2} (a))^{2} + \cos^{2} (a) \sin^{2} (a) ) - 3( \cos^{2} (a) - \sin^{2} (a) )( \cos ^{2} (a) + \sin^{2} (a) ) \\ \\ ( \cos^{2} (a) - \sin^{2} (a) )(4(( \cos^{2} (a))^{2} + ( \sin^{2} (a))^{2} + \cos^{2} (a) \sin^{2} (a) )) - 3\times 1 \\ \\ ( \cos^{2} (a) - \sin^{2} (a) )((4\cos^{4} (a) + 4\sin^{4} (a) + 4 \cos^{2} (a) \sin^{2} (a) ) - 3( \cos^{2} (a) + \sin^{2} (a) ) ^{2} ) \\ \\ ( \cos^{2} (a) - \sin^{2} (a) )((4\cos^{4} (a) + 4\sin^{4} (a) + 4 \cos^{2} (a) \sin^{2} (a) ) - 3( \cos^{4} (a) + \sin^{4} (a) + 2 \cos^{2} \sin^{2} (a) ) ) \\ \\ ( \cos^{2} (a) - \sin^{2} (a) )((cos^{4} (a) + \sin^{4} (a) - 2 \cos^{2} \sin^{2} (a) ) ) \\ \\ ( \cos^{2} (a) - \sin^{2} (a))( \cos^{2} (a) - \sin^{2} (a) ) ^{2} \\ \\ ( \cos^{2} (a) - \sin^{2} (a) )^{3} \\ \\ ( \cos(2a) )^{3} \\ \\ \cos^{3} (2a) \\ \\ = RHS \\ \\ Hence \: proved$