A) \[{{\sin }^{4}}\theta -{{\cos }^{4}}\theta \]
B) \[1-{{\sin }^{2}}\theta {{\cos }^{2}}\theta \]
C) \[1+{{\sin }^{2}}\theta {{\cos }^{2}}\theta \]
D) \[1-3{{\sin }^{2}}\theta {{\cos }^{2}}\theta \]
Correct Answer: B
Solution :
| \[\frac{{{\sin }^{6}}\theta -{{\cos }^{6}}\theta }{{{\sin }^{2}}\theta -{{\cos }^{2}}\theta }=\frac{{{({{\sin }^{2}}\theta )}^{3}}-{{({{\cos }^{2}}\theta )}^{3}}}{{{\sin }^{2}}\theta -{{\cos }^{2}}\theta }\] |
| \[=\frac{({{\sin }^{2}}\theta -{{\cos }^{2}}\theta )({{\sin }^{4}}\theta +{{\cos }^{4}}\theta +{{\sin }^{2}}\theta {{\cos }^{2}}\theta )}{{{\sin }^{2}}\theta -{{\cos }^{2}}\theta }\] |
| \[={{\sin }^{4}}\theta +{{\cos }^{4}}\theta +2{{\sin }^{2}}\theta {{\cos }^{2}}\theta -{{\sin }^{2}}\theta {{\cos }^{2}}\theta \] |
| \[={{({{\sin }^{2}}\theta +{{\cos }^{2}}\theta )}^{2}}-{{\sin }^{2}}\theta {{\cos }^{2}}\theta =1-{{\sin }^{2}}\theta {{\cos }^{2}}\theta \] |
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