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cosA-sinA+1/cosA+sinA-1
=(cosA-sinA+1)(cosA+sinA+1)/(cosA+sinA-1)(cosA+sinA+1)
=(cos²A-cosAsinA+cosA+cosAsinA-sin²A+sinA+cosA-sinA+1)/{(cosA+sinA)²-(1)²}
=(cos²A-sin²A+2cosA+1)/(cos²A+2cosAsinA+sin²A-1)
={cos²A+2cosA+(1-sin²A)}/(1+2cosAsinA-1) [∵, sin²A+cos²A=1]
=(cos²A+2cosA+cos²A)/2cosAsinA
=(2cos²A+2cosA)/2cosAsinA
=2cosA(cosA+1)/2cosAsinA
=(cosA+1)/sinA
=cosA/sinA+1/sinA
=cotA+cosecA
=cosecA+cotA (Proved)
=(cosA-sinA+1)(cosA+sinA+1)/(cosA+sinA-1)(cosA+sinA+1)
=(cos²A-cosAsinA+cosA+cosAsinA-sin²A+sinA+cosA-sinA+1)/{(cosA+sinA)²-(1)²}
=(cos²A-sin²A+2cosA+1)/(cos²A+2cosAsinA+sin²A-1)
={cos²A+2cosA+(1-sin²A)}/(1+2cosAsinA-1) [∵, sin²A+cos²A=1]
=(cos²A+2cosA+cos²A)/2cosAsinA
=(2cos²A+2cosA)/2cosAsinA
=2cosA(cosA+1)/2cosAsinA
=(cosA+1)/sinA
=cosA/sinA+1/sinA
=cotA+cosecA
=cosecA+cotA (Proved)
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