Question

\[ \text{If } \cosec x+\cot x=\frac{11}{2}, \]

\[ \text{then } \tan x= \]

(a) \(\frac{21}{22}\)
(b) \(\frac{15}{16}\)
(c) \(\frac{44}{117}\)
(d) \(\frac{117}{43}\)

Solution

Using identity

\[ (\cosec x+\cot x)(\cosec x-\cot x)=1 \]

\[ \frac{11}{2}(\cosec x-\cot x)=1 \]

\[ \cosec x-\cot x=\frac{2}{11} \]

Now,

\[ 2\cot x = (\cosec x+\cot x)-(\cosec x-\cot x) \]

\[ =\frac{11}{2}-\frac{2}{11} \]

\[ =\frac{121-4}{22} =\frac{117}{22} \]

\[ \cot x=\frac{117}{44} \]

\[ \tan x=\frac{44}{117} \]

Answer

\[ \boxed{\frac{44}{117}} \]

Correct Option: (c)