Question

Prove that :-

Cos3A-cosA/sin3A-sinA+cos2A-cos4A/sin4A-sin2A=sinA/cos2Acos3A

Answer 1

`(\cos^3A-\cos A)/(\sin^3A-\sin A)+(\cos^2A-\cos^4A)/(\sin^4A-\sin^2A)=(\sin A)/(\cos^2A\cos^3A)\\\\L_s=(\cos A(\cos^2A-1))/(\sin A(\sin^2A-1))+(\cos^2A(1-\cos^2A))/(\sin^2A(\sin^2A-1))\\\\=(\cos A(-\sin^2A))/(\sin A(-\cos^2A))+(\cos^2A\sin^2A)/(\sin^2A(-\cos^2A))\\\\=(\sin A)/(\cos A)-1=\tan A-1`


`R_s=(\sin A)/(\cos A\cos^4A)=(\sin A)/(\cos A)\cdot(1)/(\cos^4A)=\tan A\cdot(1)/(\cos^4A)=(\tan A)/(\cos^4A)`



`\boxed{L_s\\eq R_s}`