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Question

If x=secϕtanϕ and y=cosecϕ+cotϕ, then

A
x=y+1y1
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B
x=y1y+1
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C
y=1+x1x
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D
xy+xy+1=0
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Solution

The correct options are
B x=y1y+1
C y=1+x1x
D xy+xy+1=0
x=secϕtanϕ and y=cosecϕ+cotϕ
x=1sinϕcosϕ,y=1+cosϕsinϕ
Multiplying, we get
xy=(1sinϕ)(1+cosϕ)cosϕsinϕ
xy=1sinϕ+cosϕcosϕsinϕcosϕsinϕ
xy+1=1sinϕ+cosϕsinϕcosϕ+sinϕcosϕcosϕsinϕ
=1sinϕ+cosϕcosϕsinϕ
and xy=(1sinϕ)sinϕcosϕ(1+cosϕ)cosϕsinϕ
=sinϕsin2ϕcosϕcos2ϕcosϕsinϕ
=sinϕcosϕ1cosϕsinϕ=(xy+1)
Thus, xy+xy+1=0
x=y1y+1 and y=1+x1x.

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