Answer
$\frac{1+\sin 2x}{\sin 2x}=1+\frac{1}{2}\sec x\csc x$
Work Step by Step
Start with the left side:
$\frac{1+\sin 2x}{\sin 2x}$
Rewrite the expression as the sum of two fractions:
$=\frac{1}{\sin 2x}+\frac{\sin 2x}{\sin 2x}$
$=\frac{1}{\sin 2x}+1$
Use the identity $\sin 2x=2\sin x\cos x$:
$=\frac{1}{2\sin x\cos x}+1$
$=\frac{1}{2}*\frac{1}{\sin x}*\frac{1}{\cos x}+1$
Use the identities $\frac{1}{\sin x}=\csc x$ and $\frac{1}{\cos x}=\sec x$:
$=\frac{1}{2}\csc x\sec x+1$
Rearrange:
$=1+\frac{1}{2}\csc x\sec x$
$=1+\frac{1}{2}\sec x\csc x$
Since this equals the right side, the identity is proven.
