A) \[\frac{1}{4}(4+3.({{a}^{2}}-{{1}^{2}}))\]
B) \[\frac{1}{4}[4-3.({{a}^{2}}-{{1}^{2}})]\]
C) \[\frac{1}{4}[4-2.({{a}^{2}}+{{1}^{2}})]\]
D) \[\frac{1}{4}[4-3.{{({{a}^{2}}+1)}^{2}}]\]
Correct Answer: B
Solution :
[b] \[\because sinx+cosx=a\] \[2sinx.cosx={{a}^{2}}-1\] \[si{{n}^{6}}x+co{{s}^{6}}x={{(si{{n}^{2}}x)}^{3}}+{{(cos)}^{3}}\] \[={{(si{{n}^{2}}x+co{{s}^{2}}x)}^{3}}-~3.si{{n}^{2}}x.co{{s}^{2}}x(si{{n}^{2}}x+co{{s}^{2}}x)\]\[=1-3.si{{n}^{2}}x.co{{s}^{2}}x\] \[=1-3{{\left( \frac{{{a}^{2}}-1}{2} \right)}^{2}}=\frac{1}{4}[1-3{{({{a}^{2}}-1)}^{2}}]\] Hence, option [b] is correct.
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