Question

$\tiny{3 \bigg[ { \sin^{4} \bigg( \frac{3\pi}{2} - \alpha \bigg) } + \sin^{4} (3\pi - \alpha ) \bigg] - 2 \bigg [ \sin^{6} \bigg( \frac{\pi}{2} + \alpha \bigg) + { \sin}^{6} (5\pi - \alpha ) \bigg] }$​

Answer 1

Step-by-step explanation:

$3 \left[ { \sin^{4} \left( \dfrac{3\pi}{2} - \alpha \right) } + \sin^{4} (3\pi - \alpha ) \right] - 2 \left[ \sin^{6} \left( \dfrac{\pi}{2} + \alpha \right) +\sin^{6} (5\pi - \alpha ) \right]$

$= 3 \left[ { \cos^{4} \left( \alpha \right) } + \sin^{4} ( \alpha ) \right] - 2 \left[ \cos^{6} \left( \alpha \right) +\sin^{6} (\alpha ) \right]$

$= 3 \left[ \left(\cos^{2} \left( \alpha \right) + \sin^{2} ( \alpha ) \right)^{2} - 2 \sin ^{2} ( \alpha ) \cos^{2} ( \alpha ) \right] - 2 \left(\cos^{2} \left( \alpha \right) +\sin^{2} (\alpha ) \right) \left( \sin^{4} ( \alpha ) + \cos^{4} ( \alpha ) - \sin^{2} ( \alpha ) \cos^{2} ( \alpha ) \right)$

$= 3 \left[ \left(1\right)^{2} - 2 \sin ^{2} ( \alpha ) \cos^{2} ( \alpha ) \right] - 2 \left(1\right) \left( \sin^{4} ( \alpha ) + \cos^{4} ( \alpha ) - \sin^{2} ( \alpha ) \cos^{2} ( \alpha ) \right)$

$= 3 \left[ 1 - 2 \sin ^{2} ( \alpha ) \cos^{2} ( \alpha ) \right] - 2 \left \{ \left(\sin^{2} ( \alpha ) + \cos^{2} ( \alpha ) \right)^{2} - 2 \sin^{2} ( \alpha ) \cos^{2} ( \alpha ) - \sin^{2} ( \alpha ) \cos^{2} ( \alpha ) \right \}$

$= 3 - 6 \sin ^{2} ( \alpha ) \cos^{2} ( \alpha ) - 2 \left \{ \left(1 \right)^{2} - 3 \sin^{2} ( \alpha ) \cos^{2} ( \alpha ) \right \}$

$= 3 - 6 \sin ^{2} ( \alpha ) \cos^{2} ( \alpha ) - 2 \left \{ 1 - 3 \sin^{2} ( \alpha ) \cos^{2} ( \alpha ) \right \}$

$= 3 - 6 \sin ^{2} ( \alpha ) \cos^{2} ( \alpha ) - 2 + 6\sin^{2} ( \alpha ) \cos^{2} ( \alpha )$

$= 3 - 2$

$= 1$