Show that cosec(2x) + cot(2x) = cot(x)

cosec(2x) + cot(2x)

CONVERT ALL COSEC/COT/SEC FUNCTIONS INTO FUNCTIONS USING SIN/TAN/COS

= 1 / (sin2x) + cos(2x) / sin(2x)

COMBINE THE TWO FRACTIONS INTO ONE

= [1+cos(2x)] / [sin(2x)]

USE COS AND SIN DOUBLE ANGLE FORMULA

a) COS(2X) = 2COS²(X) - 1

b) SIN(2X) = 2SIN(X)COS(X)

= [1+2cos²(x)-1] / [2sin(x)cos(x)]

COLLECT LIKE TERMS

= [2cos²(x)] / [2sin(x)cos(x)]

DIVIDE BY COS(X) ON BOTH BOTTOM AND TOP OF FRACTION

= [cos(x)] / [sin(x)]

USE IDENTITY [SIN(X)] / [COS(X)] = TAN(X)

= [1] / [tan(x)]

USE IDENTITY [1] / [TAN(X)] = COT(X)

= cot(x)