If sin⁶ θ + cos⁶ θ = 1/3 , 0º θ θ cos θ ?

If sin⁶ $\theta$ + cos⁶ $\theta$ = $\frac{1}{3}$ , 0º < $\theta$ < 90º, then what is the value of sin $\theta$ cos $\theta$ ?

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Correct answer: A

Explanation

The correct answer is option 1 i.e. $\frac{\sqrt{2}}{3}$

⇒ (sin² $\theta$ )³+ (cos² $\theta$ )³ = 1/3

⇒(sin² $\theta$ + cos² $\theta$ )³− 3sin² $\theta$ cos² $\theta$ (sin² $\theta$ + cos² $\theta$ ) = 1/3

⇒(1)³− 3sin² $\theta$ cos² $\theta$ × 1 = 1/3

⇒1 − 3sin² $\theta$ cos² $\theta$ = 1/3

⇒ 3sin² $\theta$ cos² $\theta$ = 2/3

⇒ sin $\theta$ cos $\theta$ = $\frac{\sqrt{2}}{3}$ .

SSC CGL 2021 PYQ — Question 56