A) \[\cos 2\theta \]
B) \[\sin 2\theta \]
C) \[\cot 2\theta \]
D) \[\tan 2\theta \]
Correct Answer: C
Solution :
\[{{R}_{1}}=\tan \theta -{{R}_{0}}(1+\alpha {{T}_{1}})\] and \[{{R}_{2}}=\cos \theta ={{R}_{0}}(1+\alpha {{T}_{2}})\] \[\therefore \]\[\cos \theta -\tan \theta ={{R}_{0}}(1+\alpha {{T}_{2}})-{{R}_{0}}(1+\alpha {{T}_{1}})\] \[={{R}_{0}}\alpha ({{T}_{2}}-{{T}_{1}})\] or \[{{T}_{2}}-{{T}_{1}}=\frac{1}{\alpha {{R}_{0}}}(\cos \theta -\tan \theta )\] \[=\frac{1}{\alpha {{R}_{0}}}\left( \frac{\cos \theta }{\sin \theta }-\frac{\sin \theta }{\cos \theta } \right)=\frac{2\cos 2\theta }{\alpha {{R}_{0}}\sin 2\theta }\] \[=\frac{2}{\alpha {{R}_{0}}}\cos 2\theta \] Hence, the correction option is [c].
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