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Question

Prove that:

(sinA+cosecA)2+(cosA+secA)2=7+tan2A+cot2A

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Solution

LHS

(sinA+cosecA)2+(cosA+secA)2

=[sin2A+cosec2A+2sinA×cosecA]+[cos2A+sec2A+2cosA×secA]

=sin2A+cosec2A+2sinA×cosecA+cos2A+sec2A+2cosA×secA
=[sin2A+cos2A]+cosec2A+2sinA×1sinA+sec2A+2cosA×1cosA

=1+cosec2A+2+sec2A+2
=1+(1+cot2A)+2+(1+tan2A)+2[1+cot2A=coesc2A,1+tan2A=sec2A]

=7+tan2A+cot2A=RHS

LHS=RHS

(sinA+cosecA)2+(cosA+secA)2=7+tan2A+cot2A


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