First slide

Trigonometric transformations

Question

Let $f (n) = 2 \cos nx \forall n \in N, then f (1) f (n + 1) - f (n)$ is equal to

Moderate

A

f(n+3)
B

f(n+2)
C

f(n+1)f(2)
D

f(n+2)f(3)

Solution

$\begin{array}{l} f (n) = 2 \cos nx \\ \Rightarrow f (1) f (n + 1) - f (n) \\ = 4 \cos xcos (n + 1) x - 2 \cos nx \\ = 2 [2 \cos (n + 1) xcos x - \cos nx] \\ = 2 [\cos (n + 2) x + \cos nx - \cos nx] \\ = 2 \cos (n + 2) x = f (n + 2) \end{array}$

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