A) \[\cos 4\theta \]
B) \[\sin 4\theta \]
C) \[{{\sin }^{4}}\theta \]
D) \[{{\cos }^{4}}\theta \]
E) \[{{\sin }^{4}}(\theta /2)\]
Correct Answer: C
Solution :
\[\frac{1}{8}(3-4\cos 2\theta +\cos 4\theta )\] \[\Rightarrow \]\[\frac{1}{8}\{3-3\cos 2\theta +(\cos 4\theta -\cos 2\theta )\}\] \[=\frac{1}{8}\left\{ 3(1-\cos 2\theta )+2\sin \left( \frac{2\theta -4\theta }{2} \right).\sin \left( \frac{6\theta }{2} \right) \right\}\] \[=\frac{1}{8}\{6{{\sin }^{2}}\theta -2\sin \theta .\sin 3\theta \}\] \[=\frac{1}{8}\{6{{\sin }^{2}}\theta -2\sin \theta (3\sin \theta -4{{\sin }^{3}}\theta )\}\] \[=\frac{1}{8}\{6{{\sin }^{2}}\theta -6{{\sin }^{2}}\theta +8{{\sin }^{4}}\theta \}={{\sin }^{4}}\theta \]
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