if a is a rational number with ∣a∣≤1, prove that cos (n cos^(−1) (a)) is also a rational number. (n∈N)


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Question Number 79062 by mr W last updated on 22/Jan/20

$i f a i s a r a t i o n a l n u m b e r w i t h ∣ a ∣⩽ 1,$ $p r o v e t h a t \cos (n \cos^{- 1} (a)) i s a l s o a$ $r a t i o n a l n u m b e r . (n \in N)$
	Answered by mind is power last updated on 22/Jan/20

	$l e t U_{n} = c o s ({n c o s}^{-} (a))$ $U_{0} = 1 \in Q$ $U_{1} = c o s ({c o s}^{-} (a)) = a \in Q$ $s u p p o s e, \forall n ⩾ 1 U_{n}, U_{n - 1} \in Q, l e t s S h o w U_{n + 1} \in Q$ $w e W i l l u s e F o r t r e c u r s i o n$ $U_{n + 1} + U_{n - 1} = c o s ((n + 1) {c o s}^{-} (a)) + c o s ((n - 1) {c o s}^{-} (a))$ $= c o s ({n c o s}^{-} (a) + {c o s}^{-} (a)) + c o s ({n c o s}^{-} (a) - {c o s}^{-} (a))$ $= 2 a c o s ({n c o s}^{-} (a)) = 2 {a U}_{n}$ $\Rightarrow U_{n + 1} = 2 {a U}_{n} - U_{n - 1}$ $s i n c e U_{n}, U_{n - 1}, a \in Q$ $\Rightarrow U_{n + 1} = 2 {a U}_{n} - U_{n - 1} \in Q$
	Commented by mr W last updated on 22/Jan/20

	$t h a n k y o u s i r! v e r y c l e a r!$
	Commented by mind is power last updated on 22/Jan/20

	$W i t h e p l e a s u r S i r$
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