Question

prove that sin3A/sinA + cos3A/cosA = 4cos2A​

Answer 1

Solution:

$\implies \sf \dfrac{\sin 3A}{\sin A}+\dfrac{\cos 3A}{\cos A}=4\cos 2A\\ \\ \\ \underline{\bf We\;take\;LHS\;part,}\\ \\ \implies \sf \dfrac{\sin 3A}{\sin A}+\dfrac{\cos 3A}{\cos A}\\ \\ \\ \underline{\sf Now,\;take\;LCM,}\\ \\ \implies \sf \dfrac{\sin 3A\cos A+\cos 3A\sin A}{\sin A \cos A}\\ \\ \\ \underline{\sf Now,\;we\;know\;that,}\\ \\ \therefore \sf \sin a\cos b+\cos a \sin b=\sin(a+b)\\ \\ \implies \sf \dfrac{2\sin(3A+A)}{2\sin A\cos A}\\ \\ \underline{\sf We\;know\;that,}$

$\therefore \sf 2\sin A\cos A = \sin 2A\\ \\ \\ \implies \sf \dfrac{2\sin 4A}{\sin 2A}\\ \\ \\ \implies \sf \dfrac{2\times 2\sin 2A\cos 2A}{\sin 2A}\\ \\ \\ \implies \sf 4\cos 2A$

Hence Proved!!