Trigonometric identities

Identity 1: $a^6 + b^6 = \left( a^2 + b^2 \right)^3 - 3 a^2 b^2 \left(a^2 +b^2 \right)$

Identity 2: $\sin ^ 2 \theta + cos ^2 \theta =1$

Identity 3: $\sin 2 \theta = 2 \sin \theta \cos \theta$

Now using these two identities

$\begin{aligned} \sin^{6} \theta + cos^{6} \theta &= \left( \sin ^{2} \theta + \cos ^2 \theta \right)^3 - 3 sin^{2} \theta \cos^{2} \theta \left( cos^2 \theta + sin^2 \theta \right)\\ &= 1 - 3 sin^2 \theta \cos^2 \theta \\ &= 1- \dfrac{3}{4} \left(2 \sin \theta \cos \theta \right)^2 \\ &= 1- \dfrac{3}{4} \left( \sin 2 \theta \right)^2 \\ &= 1- \dfrac{3}{4} \times \left( \dfrac{2}{3} \right)^2 \\ &= 1- \dfrac{3}{4} \times \dfrac{4}{9} \\ &= 1- \dfrac{1}{3} \\ &= \dfrac{2}{3} \end{aligned}$