Chứng minh đẳng thức lượng giác:

a) sin(α + β)sin(α – β) = sin²α – sin²β;

b) cos⁴α – cos^{4 $\left(\alpha -\frac{\pi }{2}\right)$} = cos2α.

a) sin(α + β)sin(α – β) = sin²α – sin²β

Ta có: sin(α + β)sin(α – β) =  $\frac{1}{2}\left[\text{cos}\left(\alpha +\beta -\alpha +\beta \right)-\text{cos}\left(\alpha +\beta +\alpha -\beta \right)\right]$

 $={\sin }^2\alpha -{\text{sin}}^2\beta$.

b) Ta có: cos⁴α – cos^{4 $\left(\alpha -\frac{\pi }{2}\right)$} = cos⁴α – sin⁴α = (cos²α – sin²α)(cos²α + sin²α)

= cos²α – sin²α = cos2α.