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  4. The expression 3[ sin 4((3/2) π-α)+ sin 4(3 π+α)] -2[ sin 6((1/2) π+α)+ sin 6(5 π-α)] is equal to

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Q. The expression $3\left[\sin ^{4}\left(\frac{3}{2} \pi-\alpha\right)+\sin ^{4}(3 \pi+\alpha)\right]$ $-2\left[\sin ^{6}\left(\frac{1}{2} \pi+\alpha\right)+\sin ^{6}(5 \pi-\alpha)\right]$ is equal to

Trigonometric Functions

Solution:

$3\left[\sin ^{4}\left(\frac{3}{2} \pi-\alpha\right)+\sin ^{4}(3 \pi+\alpha)\right]-2$
$\left[\sin ^{6}\left(\frac{1}{2} \pi+\alpha\right)+\sin ^{6}(5 \pi-\alpha)\right]$
$=3\left(\cos ^{4} \alpha+\sin ^{4} \alpha\right)-2\left(\cos ^{6} \alpha+\sin ^{6} \alpha\right)$
$=3\left(1-2 \sin ^{2} \alpha \cos ^{2} \alpha\right)-2\left[\left(\sin ^{2} \alpha+\cos ^{2} \alpha\right)^{3}\right.$
$\left.-3 \sin ^{2} \alpha \cos ^{2} \alpha\left(\sin ^{2} \alpha+\cos ^{2} \alpha\right)\right]$
$=3\left(1-2 \sin ^{2} \alpha \cos ^{2} \alpha\right)-2\left[1-3 \sin ^{2} \alpha \cos ^{2} \alpha\right]$
$=1$