| Column - I | Column - II | ||
| P. | \[1+\frac{{{\cot }^{2}}\theta }{1+\cos ec\theta }\] | 1. | \[2\tan \theta \] |
| Q. | \[\frac{\cos \theta }{\cos ec\theta +1}\] \[+\frac{\cos \theta }{\cos ec\theta -1}\] | 2. | \[{{\left( \frac{{{\sin }^{2}}\theta -{{\cos }^{2}}\theta }{\cos \theta \,\sin \theta } \right)}^{2}}\] |
| R. | \[{{\tan }^{2}}\theta +{{\cot }^{2}}\theta -2\] | 3. | \[{{\left( \cos ec\theta -\cot \theta \right)}^{2}}\] |
| S. | \[\frac{1-\cos \theta }{1+\cos \theta }\] | 4. | \[\sec \theta \cot \theta \] |
A) P-3, Q-4, R-2, S-1
B) P-3, Q-1, R-4, S-2
C) P-2, Q-3, R-4, S-1
D) P-4, Q-1, R-2, S-3
Correct Answer: D
Solution :
| \[P\to 4;\,Q\,\to 1;\,R\to 2;\,S\to 3\] |
| (P) \[1+\frac{{{\cot }^{2}}\theta }{1+\cos ec\theta }\] |
| \[=1+\frac{\cos e{{c}^{2}}\theta -1}{1+\cos ec\theta }\] |
| \[=1+\frac{\left( \cos ec\theta \right)\left( \cos ec\theta -1 \right)}{\left( \cos ec\theta +1 \right)}\] |
| \[=1+\cos ec\theta -1\] |
| \[=\cos ec\theta =\sec \theta \cot \,\theta \] |
| (Q) \[\frac{\cos \theta }{\cos ec\theta +1}+\frac{\cos \theta }{\cos ec\theta -1}\] |
| \[=\cos \theta \left( \frac{1}{\cos ec\theta +1}+\frac{1}{\cos ec\theta -1} \right)\] |
| \[=\cos \theta \left( \frac{\cos ec\theta -1+\cos ec\theta +1}{\cos e{{c}^{2}}\theta -1} \right)\] |
| \[=\frac{\cos \theta \left( 2\cos ec\theta \right)}{{{\cot }^{2}}\theta }\] |
| \[=2\frac{\cot \theta }{{{\cot }^{2}}\theta }\] |
| \[=2\tan \theta \] |
| (R) \[{{\tan }^{2}}\theta +{{\cot }^{2}}\theta -2={{\left( \tan \theta -\cot \theta \right)}^{2}}\] |
| \[={{\left( \frac{\sin \theta }{\cos \theta }-\frac{\cos \theta }{\sin \theta } \right)}^{2}}\] |
| \[={{\left( \frac{{{\sin }^{2}}\theta -{{\cos }^{2}}\theta }{\cos \theta \sin \theta } \right)}^{2}}\] |
| (S) \[\frac{1-\cos \theta }{1+\cos \theta }=\frac{\left( 1-\cos \theta \right)\left( 1-\cos \theta \right)}{\left( 1+\cos \theta \right)\left( 1-\cos \theta \right)}\] |
| \[=\frac{{{\left( 1-\cos \theta \right)}^{2}}}{1-{{\cos }^{2}}\theta }\] |
| \[=\frac{{{\left( 1-\cos \theta \right)}^{2}}}{{{\sin }^{2}}\theta }\] |
| \[={{\left( \frac{1-\cos \theta }{\sin \theta } \right)}^{2}}\] |
| \[={{\left( \frac{1}{\sin \theta }-\frac{\cos \theta }{\sin \theta } \right)}^{2}}\] |
| \[={{\left( \cos ec\theta -\cot \theta \right)}^{2}}\] |
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