A) \[{{\sin }^{2}}\alpha +{{\sin }^{2}}\beta \]
B) \[{{\sin }^{2}}({{\alpha }^{2}}-{{\beta }^{2}})\]
C) \[{{\sin }^{2}}({{\alpha }^{2}}-{{\beta }^{2}})\]
D) \[{{\sin }^{2}}\alpha -{{\sin }^{2}}\beta \]
Correct Answer: D
Solution :
\[\sin \,\,(\alpha +\beta )\sin \,\,(\alpha -\beta )\] \[=(\sin \alpha \cos \beta +\cos \alpha \cos \beta )(\sin \alpha \cos \beta -\cos \alpha sin\beta )\]\[={{\sin }^{2}}\alpha {{\cos }^{2}}\beta -{{\cos }^{2}}\alpha {{\sin }^{2}}\beta \] \[={{\sin }^{2}}\alpha (1-{{\sin }^{2}}\beta )-(1-{{\sin }^{2}}\alpha ){{\sin }^{2}}\beta \] \[={{\sin }^{2}}\alpha -{{\sin }^{2}}\alpha {{\sin }^{2}}\beta -{{\sin }^{2}}\beta +{{\sin }^{2}}\alpha {{\sin }^{2}}\beta \] \[={{\sin }^{2}}\alpha -{{\sin }^{2}}\beta \]
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