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Question

Which of the following statements is (are) CORRECT?
(Q denotes the set of rational numbers)

A
If cos2θQ and sin2θQ, then tanθQ (if defined)
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B
If tanθQ, then sin2θ,cos2θ and tan2θQ (if defined)
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C
If sinθQ and cosθQ, then tan3θQ (if defined)
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D
If sinθQ, then cos3θQ
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Solution

The correct option is C If sinθQ and cosθQ, then tan3θQ (if defined)
tanθ=1cos2θsin2θQ
As cos2θQ and sin2θQ,
tanθQ

sin2θ=2tanθ1+tan2θ
cos2θ=1tan2θ1+tan2θ
tan2θ=2tanθ1tan2θ
If tanθQ, then we can clearly conclude from the above formulae that sin2θ,cos2θ and tan2θQ

If sinθ,cosθQ,
then sin3θ=3sinθ4sin3θQ
and cos3θ=4cos3θ3cosθQ
tan3θ=sin3θcos3θQ

If sinθQ,
then sin3θ=3sinθ4sin3θQ
But cos3θ=±1sin23θ need not to be rational.
For e.g. sinθ=13Q
sin3θ=3×134×(13)3=2327
But, cos3θ=±20027Q

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