Prove that cos⁡ec2A⋅sec2A=tan2A+cot2A+2

Alex Cervantes

Alex Cervantes

Answered

2022-01-24

Prove that cosec2Asec2A=tan2A+cot2A+2

Answer & Explanation

Tapanuiwp

Tapanuiwp

Expert

2022-01-25Added 13 answers

I assume your second csc2A was meant to be sec2A, viz.
tan2A+cot2A+2=(tanA+cotA)2=(sin2A+cos2AsinAcosA)2
=1sin2Acos2A=csc2Asec2A
pripravyf

pripravyf

Expert

2022-01-26Added 12 answers

Let s=sinA  and  c=cosA. Then tanA=sc and cotA=cs. Therefore
RHS=tan2A+cot2A+2=(sc)2+(cs)2+2
This gives
RHS=s4+c4+2s2c2s2c2=(s2+c2)2s2c2
But s2+c2=sin2A+cos2A=1. Therefore
RHS=1s2c2=(1s)2(1c)2
Since cosecA=1s and secA=1c, the claim follows (presumably you wanted to prove cosec2Asec2A=tan2A+cot2A+2)

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