Question

${\sin }^2\alpha +{\cos }^2(\alpha +\beta )+2\sin \alpha .\sin \beta \cos (\alpha +\beta )=$

• A. $si{n}^2\alpha$
• B. $si{n}^2\beta$
• C. $co{s}^2\alpha$
• D. $co{s}^2\beta$

Answer 1

The correct option is D $co{s}^2\beta$

Given expression is ${\sin }^2\alpha +\cos (\alpha +\beta )\left[\cos \right(\alpha +\beta )+2\sin \alpha \sin \beta ]$
= ${\sin }^2\alpha +\cos (\alpha +\beta )[\cos \alpha \cos \beta -\sin \alpha \sin \beta +2\sin \alpha \sin \beta ]$
= ${\sin }^2\alpha +\cos (\alpha +\beta )\left[\cos \right(\alpha -\beta \left)\right]={\sin }^2\alpha +{\cos }^2\beta -{\sin }^2\alpha ={\cos }^2\beta$