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Step-by-step explanation:

LHS = (tan β + Sec β)(1 - sin β)

= ((sin β / cos β) + (1/cos β))(1-sin β)

= ((1+ sin β)*(1- sin β))/cos β

= (1 - sin^2 β) / cos β

= cos^2 β / cos β

= cos β

= RHS

Hence proved