A) \[\frac{{{\sin }^{3}}(\alpha +\beta )}{\cos \alpha }\]
B) \[\cos (\alpha +3\beta )\]
C) 0
D) None of these
Correct Answer: C
Solution :
\[y={{\sin }^{2}}\alpha +{{\cos }^{2}}(\alpha +\beta )+2\sin \alpha \sin \beta \cos (\alpha +\beta )\] \[={{\sin }^{2}}\alpha +\cos (\alpha +\beta )\{\cos (\alpha +\beta )+2\sin \alpha \sin \beta \}\] \[={{\sin }^{2}}\alpha +\cos (\alpha +\beta )\cos (\alpha -\beta )\] \[={{\sin }^{2}}\alpha +\frac{1}{2}(\cos 2\alpha +\cos 2\beta )\] \[={{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha -\frac{1}{2}+\frac{\cos 2\beta }{2}\] Þ \[y=\]constant Þ \[\frac{{{d}^{3}}y}{d{{\alpha }^{3}}}=0\] Trick: Let \[\beta =180{}^\circ \]{since \[\beta \]is constant} \[\therefore y={{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha =1\Rightarrow \frac{{{d}^{3}}y}{d{{\alpha }^{3}}}=0\].
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