Question

\[ \text{If } \cosec x-\cot x=\frac12,\ 0<x<\frac{\pi}{2}, \]

\[ \text{then } \cos x \text{ is equal to} \]

(a) \(\frac53\)
(b) \(\frac35\)
(c) \(-\frac35\)
(d) \(-\frac53\)

Solution

Using identity

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

\[ \frac12(\cosec x+\cot x)=1 \]

\[ \cosec x+\cot x=2 \]

Now,

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

\[ =2+\frac12 =\frac52 \]

\[ \cosec x=\frac54 \]

\[ \sin x=\frac45 \]

Since \(0<x<\frac{\pi}{2}\),

\[ \cos x=\sqrt{1-\sin^2x} \]

\[ =\sqrt{1-\frac{16}{25}} \]

\[ =\frac35 \]

Answer

\[ \boxed{\frac35} \]

Correct Option: (b)