First slide

Trigonometric equations

Question

One of the general solutions of $4 \sin^{4} x + \cos^{4} x = 1$ is

Moderate

A

$nπ \pm α / 2, α = \cos^{- 1} (1 / 5), \forall n \in Z$
B

$nπ \pm α / 2, α = \cos^{- 1} (3 / 5), \forall n \in Z$
C

$2 nπ \pm α / 2, α = \cos^{- 1} (1 / 3), \forall n \in Z$
D

none of these

Solution

$\begin{array}{l} 4 \sin^{4} x + \cos^{4} x = 1 \\ \Rightarrow {(2 \sin^{2} x)}^{2} + \frac{1}{4} {(2 \cos^{2} x)}^{2} = 1 \end{array}$
$\begin{array}{l} or (1 - \cos 2 x)^{2} + \frac{1}{4} (1 + \cos 2 x)^{2} = 1 \\ or 5 \cos^{2} 2 x - 6 \cos 2 x + 1 = 0 \\ or (\cos 2 x - 1) (5 \cos 2 x - 1) = 0 \\ or \cos 2 x = 1 or \cos 2 x = \frac{1}{5} \\ \Rightarrow 2 x = 2 nπ or 2 x = 2 nπ \pm α \end{array}$
where $α = \cos^{- 1} (\frac{1}{5}), \forall n \in Z$

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