| The value of is \[\cot \theta \cdot \tan \,(90{}^\circ -\theta )-sec\,(90{}^\circ -\theta )\]\[\text{cosec}\theta +({{\sin }^{2}}25{}^\circ +{{\sin }^{2}}65{}^\circ )+\] \[\sqrt{3}\,(\tan 5{}^\circ \cdot \tan 15{}^\circ \cdot \tan 30{}^\circ \cdot \tan 75{}^\circ \cdot \tan 85{}^\circ )\]is [SSC (10+2) 2012] |
A) 1
B) \[-1\]
C) 2
D) 0
Correct Answer: A
Solution :
| Expression = |
| \[=\cot \theta \cdot \tan \,(90{}^\circ -\theta )-sec\,(90{}^\circ -\theta )\] |
| \[\text{cosec}\theta +({{\sin }^{2}}25{}^\circ +{{\sin }^{2}}65{}^\circ )+\sqrt{3}\] |
| \[(\tan 5{}^\circ \cdot tan15{}^\circ \cdot \tan 30{}^\circ \cdot \tan 75{}^\circ \cdot \tan 85{}^\circ )\] |
| \[=\cot \theta \cdot \cot \theta -\text{cosec}\theta \cdot \text{cosec}\theta +\] |
| \[({{\sin }^{2}}25{}^\circ +{{\cos }^{2}}25{}^\circ )+\sqrt{3}\,\,(\tan 5{}^\circ \] |
| \[\cot 5{}^\circ \cdot \tan 15{}^\circ \cot 15{}^\circ \cdot \tan 30{}^\circ )\] |
| \[\left[ \begin{matrix} \because \sec \,(90{}^\circ -\theta )=\text{cosec}\theta \\ \sin \,(90{}^\circ -\theta )=\cos \theta \\ \tan \,(90{}^\circ -\theta )=\cot \theta \\ \end{matrix} \right]\] |
| \[=({{\cot }^{2}}\theta -\text{cose}{{\text{c}}^{2}}\theta )+({{\sin }^{2}}25{}^\circ +{{\cos }^{2}}25{}^\circ )+\sqrt{3}\times \frac{1}{\sqrt{3}}\]\[=-1+1+1=1\] |
| \[[\because {{\sin }^{2}}\theta {{\cos }^{2}}\theta =1\,\text{and}\,{{\cot }^{2}}\theta -\text{cose}{{\text{c}}^{2}}\theta =-1]\] |
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