Question

\[ \text{If } \cosec x+\cot x=\alpha, \]

\[ \text{then } \sin x=\ ? \]

Solution

Using identity

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

\[ \cosec x-\cot x=\frac1\alpha \]

Adding,

\[ 2\cosec x = \alpha+\frac1\alpha \]

\[ \cosec x = \frac{\alpha^2+1}{2\alpha} \]

Therefore,

\[ \sin x = \frac{2\alpha}{\alpha^2+1} \]

Answer

\[ \boxed{\frac{2\alpha}{\alpha^2+1}} \]