Question

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

• A. $0$
• B. $1$
• C. $3$
• D. $\sin 4\alpha +\cos 6\alpha$

Answer 1

Answer: (B)

$1$

$3[co{s}^4\alpha +si{n}^4\alpha ]-2[co{s}^6\alpha +si{n}^6\alpha ]$

$=3\left[\right(co{s}^2\alpha {)}^2+\left(si{n}^2{\alpha }^2\right)]-2[(co{s}^2\alpha {)}^3+(si{n}^2\alpha {)}^3]$

$=3[1-\frac{1}{2}si{n}^{2}2\alpha ]-2[1-\frac{3}{4}si{n}^{2}2\alpha ]$

$=3-2$

$=1$