A) 1
B) zero
C) 2
D) 3
Correct Answer: B
Solution :
\[{{P}_{n}}={{\cos }^{n}}\theta +{{\sin }^{n}}\theta \] when \[n=6\Rightarrow {{P}_{6}}={{\cos }^{6}}\theta +{{\sin }^{6}}\theta \] \[n=4\Rightarrow {{P}_{4}}={{\cos }^{4}}\theta +{{\sin }^{4}}\theta \] \[=2(co{{s}^{6}}\theta +si{{n}^{6}}\theta )-3(co{{s}^{4}}\theta +si{{n}^{4}}\theta )+1\] \[=2\{(co{{s}^{2}}\theta +si{{n}^{2}}\theta )\] \[(co{{s}^{4}}\theta +{{\sin }^{4}}\theta -{{\sin }^{2}}\theta co{{s}^{2}}\theta \}-\] \[3\{co{{s}^{4}}+si{{n}^{4}}\theta \}+1\] \[=2{{\cos }^{2}}\theta +2{{\sin }^{4}}\theta -{{\sin }^{2}}\theta {{\cos }^{2}}\theta \] \[-3{{\cos }^{4}}\theta -3{{\sin }^{4}}\theta +1\] \[=-{{\cos }^{4}}\theta -{{\sin }^{4}}\theta -2{{\sin }^{2}}\theta {{\cos }^{2}}\theta +1\] \[=-(co{{s}^{4}}\theta +si{{n}^{4}}\theta +2si{{n}^{2}}\theta {{\cos }^{2}}\theta )+1\] \[=-1+1=0\]
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