| If \[\tan 15{}^\circ =2-\sqrt{3},\] then the value of\[\tan 15{}^\circ \cot 75{}^\circ +\tan 75{}^\circ \cot 15{}^\circ \]is [SSC (10+2) 2011] |
A) 14
B) 12
C) 10
D) 8
Correct Answer: A
Solution :
| Given, \[\tan 15{}^\circ =2-\sqrt{3}\] |
| Then, \[\tan 15{}^\circ \cdot \cot 75{}^\circ +tan75{}^\circ \cdot cot15{}^\circ \] |
| \[=\tan 15{}^\circ \cdot \cot \,\,(90{}^\circ -15{}^\circ )+\tan \,\,(90{}^\circ -15{}^\circ )\cdot \cot 15{}^\circ \]\[={{\tan }^{2}}15{}^\circ +{{\cot }^{2}}15{}^\circ \] (i) |
| \[[\because \tan \,\,(90{}^\circ -\theta )=\cot \theta ,\cot \,\,(90{}^\circ -\theta )=\tan \theta ]\] |
| Now, \[\tan 15{}^\circ =2-\sqrt{3}\] |
| \[\Rightarrow \]\[\cot 15{}^\circ =\frac{1}{2-\sqrt{3}}=\frac{2+\sqrt{3}}{(2-\sqrt{3})(2+\sqrt{3})}\] |
| \[=2+\sqrt{3}\] [on rationalisation] |
| \[\therefore \]\[{{\tan }^{2}}15{}^\circ +{{\cot }^{2}}15{}^\circ ={{(2-\sqrt{3})}^{2}}+{{(2+\sqrt{3})}^{2}}\] |
| \[=[4+3-4\sqrt{3}]+[4+3+4\sqrt{3}]\] |
| \[=7+7=14\] |
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