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cos a + 2 cos 2a + cos 3a/sin a+2sin2a + sin 3 a

Sin2A = 2sinAcosA; cos2A = 2cos^2A - 1

sin3A = sin (A+2A) = sinAcos2A + cosAsin2A = sinA (2cos^2A-1) + cosA (2sinAcosA)

= 2sinAcos^2A - sinA + 2sinAcos^2A

cos3A = cos (A+2A) = cosAcos2A - sinAsin2A = cosA (2cos^2A-1) - sinA (2sinAcosA)

= 2cos^3A-cosA - 2sin^2AcosA

Hence the left side of your equation equals

(2sinAcosA+4sinAcos^2A) / (2cos^2A - 1 + 2cos^3A - 2sin^2AcosA), now replace sin^2A by 1-cos^2A

= (2sinAcosA+4sinAcos^2A) / (4cos^3A + 2cos^2A - 2cosA - 1)

= 2sinAcosA (1+2cosA) / ((2cos^2A-1) (1+2cosA))

= 2sinAcosA / (2cos^2A - 1)

= sin2A / cos2A

= tan2A