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