A) \[4\sin \left( \frac{\lambda }{2} \right)\]
B) \[4\cos \left( \frac{\lambda }{2} \right)\]
C) \[2\sqrt{2}sin\left( \frac{\lambda }{2}+\frac{\pi }{4} \right)\]
D) \[2\sqrt{2}sin\left( \lambda +\frac{\pi }{4} \right)\]
Correct Answer: C
Solution :
[c] \[\sin \alpha +\sin \beta +cos\alpha +cos\beta \] \[=2\sin \left( \frac{\alpha +\beta }{2} \right)\cos \left( \frac{\alpha -\beta }{2} \right)+2\cos \left( \frac{\alpha +\beta }{2} \right)\cos \left( \frac{\alpha -\beta }{2} \right)\]\[=4\cos \left( \frac{\alpha -\beta }{2} \right)\,\left[ \sin \frac{\lambda }{2}+\cos \frac{\lambda }{2} \right]\] \[\le 2\left[ \sin \frac{\lambda }{2}+\cos \frac{\lambda }{2} \right]\] \[=2\sqrt{2}\sin \left( \frac{\lambda }{2}+\frac{\pi }{4} \right)\] (when \[\alpha =\beta =\lambda /2\])
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