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If $0 \leq x \leq π$ and $81^{\sin^{2} x} + 81^{\cos^{2} x} = 30$ , then x is equal to

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a

π6,π3

b

π3,π2

c

5π6,π3

d

2π3,π3

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detailed solution

Correct option is A

Let 81sin2⁡x=y. Then, 81cos2⁡x=811−sin2⁡x=81y−1Now, 81sin2⁡x+81cos2⁡x=30⇒y+81y=30⇒y2−30y+81=0⇒ y=3 or, y=27⇒ 81sin2⁡x=3 or 81sin2⁡x=27=34sin2⁡x=31 or 34sin2⁡x=33⇒ 4sin2⁡x=1,4sin2⁡x=3⇒ sin⁡x=±12 or, sin⁡x=±32⇒x=π6 or π3

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