Question

If ${\sin }^4\alpha +4{\cos }^4\beta +2=4\sqrt{2}$ $\sin \alpha$ $\cos \beta ;\alpha ,\beta ϵ[0,\pi ]$, then $\cos (\alpha +\beta )-\cos (\alpha +\beta )$ is equal to :

• A. $0$
• B. $-\sqrt{2}$
• C. $-1$
• D. $\sqrt{2}$

Answer 1

Answer: (B)

$-\sqrt{2}$

A.M. $\ge$ G.M.

$\frac{{\sin }^4\alpha +4{\cos }^4\beta +1+1}{4}\ge ({\sin }^4\alpha .4.{\cos }^4\beta .1.1{)}^{\frac{1}{4}}$

$\Rightarrow A.M.=G.M.\Rightarrow {\sin }^4\alpha =1=4{\cos }^4\beta$

$\sin \alpha =1,\cos \beta =\pm \frac{1}{\sqrt{2}}$

$\sin \beta =\frac{1}{\sqrt{2}}$ as $\beta ϵ[0,\pi ]$

$\cos (\alpha +\beta )-\cos (\alpha -\beta )=-2\sin \alpha \sin \beta$

= $-\sqrt{2}$