| I. \[\cos 2\alpha +\cos 2\beta +\cos 2\gamma =-1\] |
| II. \[{{\sin }^{2}}\alpha +{{\sin }^{2}}\beta +{{\sin }^{2}}\gamma =1\] |
| Which of the above is/are correct? |
A) Only I
B) Only II
C) Both I and II
D) Neither I nor II
Correct Answer: A
Solution :
[a] We have. \[{{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\cos }^{2}}\gamma =1\] ??(i) \[\Rightarrow 2{{\cos }^{2}}\alpha +2{{\cos }^{2}}\beta +2{{\cos }^{2}}\gamma =2\] \[\Rightarrow 2{{\cos }^{2}}\alpha -1+2{{\cos }^{2}}\beta -1+2{{\cos }^{2}}\gamma -1=2-3\] \[\Rightarrow \cos 2\alpha +\cos 2\beta +\cos 2\gamma =-1\] Hence statement- I is correct. And now from (i), \[1-{{\sin }^{2}}\alpha +1-{{\sin }^{2}}\beta +1-{{\sin }^{2}}\gamma =1\] \[\Rightarrow {{\sin }^{2}}\alpha +{{\sin }^{2}}\beta +{{\sin }^{2}}\gamma =2\] Hence, only statement I is correct.
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