If f x = 2 tan   3 x + 5 1 - cos   6 x ;   g x is a function having the same time period as that of f(x), then which of the following can be g(x).
(a) sec 2 3 x + cosec 2 3 x tan 2 3 x                               (b) 2   sin 3 x + 3 cos 3 x
(c) 2 1 - cos 2 3 x + cosec 3 x                                 (d) 3   cosec 3 x + 2   tan 3 x

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Please find below the solution to the asked query:

Note: If period of fx is T, then period of fax is Tafx=2tan3x+51-cos6x=2tan3x+51-1+2sin23x  As cos2A=1-2sin2A=2tan3x+52sin23xHencefx=2tan3x+52sin3xNow period of sinx is π, hence that of sin3x is π3Period of tanx is π, hence that of tan3x is π3HencePeriod of fx=L.C.M.π3,π3=π3Hence optionA is correct becausePeriod of sec2x, cosec2x and tan2x is π, hence that ofsec23x+cosec23x tan23x will be  π3

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