A) \[2\sin \theta \,\,.\,\,\cos \theta \]
B) \[\operatorname{cosec}\theta \,\,.\,\,sec\theta \]
C) \[\sin \,\theta \,\,.\,\,cos\theta \]
D) \[2cosec\,\theta \,\,.\,\,sin\theta \]
Correct Answer: C
Solution :
\[\frac{\tan \theta }{{{(1+{{\tan }^{2}}\theta )}^{2}}}+\frac{\cot \theta }{{{(1+{{\cot }^{2}}\theta )}^{2}}}=\frac{\tan \theta }{se{{c}^{4}}\theta }+\frac{\cot \theta }{\text{cose}{{\text{c}}^{4}}\theta }\] \[[\because \,{{\sec }^{2}}\theta =1+{{\tan }^{2}}\theta ,\text{cose}{{\text{c}}^{2}}\theta =1+{{\cot }^{2}}\theta ]\] \[=\sin \theta \,{{\cos }^{3}}\theta +\cos \theta {{\sin }^{3}}\theta \] \[=\sin \theta \cos \theta ({{\cos }^{2}}\theta +{{\sin }^{2}}\theta )\] \[=\sin \theta \,\cos \theta \]
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