A) 0
B) \[-1\]
C) \[\sqrt{2}\]
D) \[-\sqrt{2}\]
Correct Answer: D
Solution :
| \[A.M.\le \,G.M.\] |
| \[\frac{{{\sin }^{4}}\alpha +4{{\cos }^{4}}\beta +1+1}{4}\ge {{\left( {{\sin }^{4}}\alpha .4{{\cos }^{4}}\beta .1.1 \right)}^{\frac{1}{4}}}\] |
| \[{{\sin }^{4}}\alpha +4{{\cos }^{4}}\beta +2=4\sqrt{2}\sin \alpha \,\,\cos \beta \] |
| given that, |
| \[\Rightarrow \]A.M = G.M |
| \[\Rightarrow \]\[{{\sin }^{4}}\alpha =\pm 1\] |
| \[\cos \beta =\pm \frac{1}{\sqrt{2}},\]As \[\alpha ,\beta \in [0,\pi ]\] |
| \[\Rightarrow \]\[\sin \alpha =1,\cos \beta =\pm \frac{1}{\sqrt{2}}\]\[\Rightarrow \]\[\sin \beta =\frac{1}{\sqrt{2}}\]as \[\beta \in [0,\pi ]\] |
| \[\cos \,(\alpha +\beta )-\cos \,(\alpha -\beta )=-2\sin \alpha \,\sin \beta \]\[=-\sqrt{2}.\] |
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