A) \[\pi \]
B) \[\pi /2\]
C) \[\pi /3\]
D) \[\pi /4\]
Correct Answer: D
Solution :
We know that \[\sin \,(A+B)=\sin A\cos B+\cos A\sin B\] \[=\frac{1}{\sqrt{10}}\sqrt{1-\frac{1}{5}}+\frac{1}{\sqrt{5}}\,\sqrt{1-\frac{1}{10}}\] \[=\frac{1}{\sqrt{10}}\sqrt{\frac{4}{5}}+\frac{1}{\sqrt{5}}\sqrt{\frac{9}{10}}=\frac{1}{\sqrt{50}}(2+3)=\frac{5}{\sqrt{50}}=\frac{1}{\sqrt{2}}\] \[\Rightarrow \,\,\sin \,(A+B)=\sin \frac{\pi }{4}\] Hence, \[A+B=\frac{\pi }{4}\].
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