if-xsina-cosa-n-q-x-2-1-r-find-r-

Question Number 73056 by mathmax by abdo last updated on 05/Nov/19

i f {(x s i n a + c o s a)}^{n} = q (x^{2} + 1) + r f i n d r

Commented by mathmax by abdo last updated on 06/Nov/19

d e g r < 2 \Rightarrow r (x) = α x + β \Rightarrow {(c o s a + x s i n a)}^{n} = q (x^{2} + 1) + α x + β

x = i \Rightarrow {(c o s a + i s i n a)}^{n} = α i + β \Rightarrow c o s (n a) + i s i n (n a) = α i + β

x = - i \Rightarrow {(c o s a - i s i n a)}^{n} = - α i + β \Rightarrow c o s (n a) - i s i n (a) = - α i + β \Rightarrow

\Rightarrow 2 c o s (n a) = 2 β \Rightarrow β = c o s (n a) a n d 2 i s i n (n a) = 2 α i \Rightarrow

α = s i n (n a) \Rightarrow r (x) = s i n (n a) x + c o s (n a)