Question

\[ \text{If } \frac{3\pi}{4}<x<\pi, \]

\[ \sqrt{\cosec^2x+2\cot x} \]

\[ \text{is equal to} \]

Solution

Using identity

\[ \cosec^2x=1+\cot^2x \]

Therefore,

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

\[ =1+\cot^2x+2\cot x \]

\[ =(\cot x+1)^2 \]

Hence,

\[ \sqrt{\cosec^2x+2\cot x} = |\cot x+1| \]

Since

\[ \frac{3\pi}{4}<x<\pi \]

\(x\) lies in second quadrant and

\[ -1<\cot x<0 \]

So,

\[ \cot x+1>0 \]

Therefore,

\[ |\cot x+1|=\cot x+1 \]

Answer

\[ \boxed{1+\cot x} \]