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Question
If \[cosec x + \cot x = \frac{11}{2}\], then tan x =
Options
- \[\frac{21}{22}\]
- \[\frac{15}{16}\]
- \[\frac{44}{117}\]
- \[\frac{117}{43}\]
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Solution
We have:
\[ cosec x + \cot x = \frac{11}{2} \left( 1 \right)\]
\[ \Rightarrow \frac{1}{cosec x + \cot x} = \frac{2}{11}\]
\[ \Rightarrow \frac{{cosec}^2 x - \cot^2 x}{cosec x + \cot x} = \frac{2}{11}$\]
\[ \Rightarrow \frac{\left( cosec x + \cot x \right)\left( cosec x - \cot x \right)}{\left( cosec x + \cot x \right)} = \frac{2}{11}\]
\[ \therefore cosecx-\cot x = \frac{2}{11} \left( 2 \right)\]
Subtracting ( 2 ) from (1):
\[2\cot x = \frac{11}{2} - \frac{2}{11}\]
\[ \Rightarrow 2\cot x = \frac{121 - 4}{22}\]
\[ \Rightarrow 2\cot x = \frac{117}{22}\]
\[ \Rightarrow \cot x=\frac{117}{44}\]
\[ \Rightarrow \frac{1}{\tan x} = \frac{117}{44}\]
\[ \Rightarrow \tan x = \frac{44}{117}\]
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