Question

Help!!!!!!!!!!!!!!!!​

Answer 1

`\cos\alpha\sin\alpha(\cos\alpha-\sin\alpha)`

Explanation:

First, simplify each term:

`\sin\left((\pi)/(2)+\alpha\right)=\cos \alpha\\ \\\cos \left((\pi)/(2)+\alpha\right)=-\sin \alpha\\ \\\cos \left(\alpha-(3\pi)/(2)\right)=-\sin \alpha\\ \\\sin \left((3\pi)/(2)+\alpha\right)=-\cos \alpha`

Then given expression is equivalent to

`\cos ^3\alpha+(-\sin \alpha)^3-(-\sin \alpha)+(-\cos \alpha)\\ \\=\cos ^3\alpha-\sin^3 \alpha+\sin \alpha-\cos \alpha\\ \\=(\cos\alpha-\sin\alpha)(\cos^2\alpha+\cos\alpha\sin\alpha+\sin^2\alpha)-(\cos\alpha-\sin\alpha)\\ \\=(\cos\alpha-\sin\alpha)(1+\cos\alpha\sin\alpha-1)\ \ [\cos^2\alpha+\sin^2\alpha=1]\\ \\=\cos\alpha\sin\alpha(\cos\alpha-\sin\alpha)`