A) \[2\sin 2\theta \]
B) \[2\cos 2\theta \]
C) \[\tan 2\theta \]
D) \[\cot 2\theta \]
Correct Answer: A
Solution :
Let \[\frac{\sin 3\theta -\cos 3\theta }{\sin \theta +\cos \theta }=\frac{N}{D}\](say) Then \[N=3\sin \theta -4{{\sin }^{3}}\theta -(4{{\cos }^{3}}\theta -3\cos \theta )\] \[=3(\sin \theta +\cos \theta )-4({{\sin }^{3}}\theta +{{\cos }^{3}}\theta )\] \[=(\sin \theta +\cos \theta )\{3-4({{\sin }^{2}}\theta -\sin \theta \cos \theta +{{\cos }^{2}}\theta )\}\] \[\therefore \ \frac{N}{D}+1=\]\[\frac{(\sin \theta +\cos \theta )\{3-4(1-\sin \theta \cos \theta )\}}{\sin \theta +\cos \theta }+1\] \[=3-4(1-\sin \theta \cos \theta )+1\]\[=4\sin \theta \cos \theta =2\sin 2\theta \].
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