Question

\[ \text{If } \sin x+\cosec x=2, \]

\[ \text{then } \sin^2x+\cosec^2x= \]

Solution

Let

\[ a=\sin x \]

Then

\[ a+\frac1a=2 \]

Squaring both sides,

\[ \left(a+\frac1a\right)^2=4 \]

\[ a^2+\frac1{a^2}+2=4 \]

Therefore,

\[ a^2+\frac1{a^2}=2 \]

\[ \sin^2x+\cosec^2x=2 \]

Answer

\[ \boxed{2} \]