Which of the following statements are always correct (where Q denotes the set of rationals)? `cos2theta in Qa n dsin2theta in Q tantheta in Q` `(ifd efin e d)` `tantheta in Q sin2theta,cos2thetaa n dtan2theta in Q(ifd efin e d)` `ifs intheta in Qa n dcostheta in Qtan3theta in Q(ifd efin e d)` fi sin `theta in Q cos3theta in Q`

Options

A. `cos 2theta in Q` and `sin 2 theta in QrArr tan theta in Q` (if defined)
B. `tan theta in QrArr sin 2theta, cos 2theta` and `tan 2theta in Q` (if denied)
C. if `sinthetain Q` and `cos theta inQrArr tan 3thetainQ`(if denied)
D. if `sin theta in QrArr cos 3theta inQ`

Answer

A, B, C

Explanation

(1) `tan theta=(1-cos 2theta)/(sin 2theta)` Hence option 1 is correct.
(2) `sin 2theta=(2tan theta)/(1+tan^(2)theta)`
`cos 2theta=(1-tan^(2)theta)/(1+tan^(2)theta)`
`tan 2theta=(2tan theta)/(1-tan^(2)theta)`
Hence, option (2) is correct.
(3) `tan 3theta=(sin 3theta)/(cos 3theta)` Hence, option (3) is correct.
(4) `sin theta=1//3`, which is rational but `cos 3theta=costheta(4cos^(2)theta-3)`.
Which is irrational
Hence, option (4) is incorrect.