3 [ ^4 (3 2 - )+ .& . ^4(3 + ) ]-2 [ ^6 ( 2 + )+ ^6(5 - ) ]= - Vidyadip

MCQ

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

A
$0$
✓
1
C
3
D
$\sin4\alpha+\sin6\alpha$

✓

Answer

Correct option: B.

1

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

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