Prove that 2(cosec4A+cot4A) = cotA - tanA

  2 (cosec4A + cot4A )  = 2 ( 1/ sin4A + cos4A / sin4A )

  =  2 (  1+ cos4A / sin4A )

  = 2 (  2 cos2 2A / 2 sin2A cos2A )

  = 2(  cos2A / sin2A )

  = 2( cos2 A - sinA / 2 sinA cosA )

  =  cotA - tanA

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L.H.S = 2(cosec4A + cot4A)

= 2( (1 / sin4A) + (cos4A / sin4A) )

= 2( (1 + cos4A) / sin4A)

= 2( (1 + cos2(2A) ) / sin2(2A) )

= 2( (1 + cos22A - sin22A ) / 2sin2Acos2A )

= 2( (cos22A + cos22A) / 2sin2Acos2A )

= 2( 2cos22A / 2sin2Acos2A )

= 2( cos2A / sin2A )

= 2( ( cos2A - sin2A ) / 2sinAcosA)

= ( cos2A / sinAcosA ) - ( sin2A / sinAcosA)

= ( cosA / sinA ) - ( sinA / cosA )

= cotA - tanA

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