If 4 ^4 x + ^4 x = 1, then x = - Vidyadip

MCQ

If $4{\sin ^4}x + {\cos ^4}x = 1,$ then $x =$

✓
$n\pi $
B
$n\pi \pm {\sin ^{ - 1}}\frac{2}{5}$
C
$n\pi + \frac{\pi }{6}$
D
None of these

✓

Answer

Correct option: A.

$n\pi $

a
(a) The given equation can be put in the form

$4{\sin ^4}x = 1 - {\cos ^4}x = (1 - {\cos ^2}x)\,(1 + {\cos ^2}x)$

$ \Rightarrow $ ${\sin ^2}x[4{\sin ^2}x - 1 - (1 - {\sin ^2}x)] = 0$

$ \Rightarrow $${\sin ^2}x[5{\sin ^2}x - 2] = 0$

$ \Rightarrow $$\sin x = 0$ or $\sin x = \pm \sqrt {2/5} $.

Hence $x = n\pi $ is the required answer.

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