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Question

Prove that cosA-sinA+1cosA+sinA-1=cosecA+cotA


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Solution

Solve for the required proof

Given that cosA-sinA+1cosA+sinA-1=cosecA+cotA

Consider L.H.S

cosA-sinA+1cosA+sinA-1

Dividing the numerator and denominator by sinA, we get,

cosAsinA-sinAsinA+1sinAcosAsinA+sinAsinA-1sinA

=cotA-1+cosecAcotA+1-cosecA ; cotA=cosAsinA,cosecA=1sinA

=cotA-cosec2A-cot2A+cosecAcotA+1-cosecA ;1+cot2A=cosec2A

=cotA+cosecA-cosecA+cotAcosecA-cotAcotA+1-cosecA ; a2-b2=a+ba-b

=cotA+cosecA1-cosecA-cotAcotA+1-cosecA

=cotA+cosecAcotA+1-cosecAcotA+1-cosecA

=cotA+cosecA

=R.H.S

L.H.S=R.H.S

Hence, the given identity is proved.


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