| Assertion If \[\tan \theta +\cot \theta =2,\] then \[{{\tan }^{n}}\theta +{{\cot }^{n}}\theta =2\] for all \[n\in N.\] |
| Reason (R) \[\tan \theta +\cot \theta ={{\tan }^{n}}\theta +{{\cot }^{n}}\theta \]for all \[n\in N.\] |
A) A and R are correct and R is correct explanation of A
B) A and R are correct but R is not correct explanation of A
C) A is correct but R is wrong
D) A is wrong but R is correct
Correct Answer: A
Solution :
| Given, \[\tan \theta +\cot \theta =2\] |
| \[\Rightarrow \] \[\tan \theta +\frac{1}{\tan \theta }=2\] |
| \[\Rightarrow \] \[{{\tan }^{2}}\theta -2\tan \theta +1=0\] |
| \[\Rightarrow \] \[{{(\tan \theta -1)}^{2}}=0\] |
| \[\Rightarrow \] \[\tan \theta =1\] |
| \[\therefore \] \[{{\tan }^{n}}\theta +{{\cot }^{n}}\theta =1+1=2\] |
| \[\therefore \] A and R are individually true and R is the correct explanation of A. |
| Shortcut method |
| We know that, |
| \[AM\ge GM\] |
| \[\Rightarrow \] \[\frac{{{\tan }^{n}}\theta +{{\cot }^{n}}\theta }{2}\ge {{({{\tan }^{n}}\theta \cdot {{\cot }^{n}}\theta )}^{\frac{1}{2}}}\] |
| \[\Rightarrow \] \[{{\tan }^{n}}\theta +{{\cot }^{n}}\theta \ge 2,\]\[=\sqrt{1+2+1-2\sqrt{2}}=\sqrt{4-2\sqrt{2}}\] |
| \[\therefore \] \[\tan \theta +\cot \theta \ge 2,\]when\[n=1\] |
You need to login to perform this action.
You will be redirected in
3 sec
