Answer
$$\frac{1+\cos2x}{\sin2x}=\cot x$$
The equation is an identity, because the two sides are equal.
Work Step by Step
$$\frac{1+\cos2x}{\sin2x}=\cot x$$
We need to prove that the left side is equal to the right one.
$$X=\frac{1+\cos2x}{\sin2x}$$
From Double-Angle Identities, we have
$$\cos2x=2\cos^2x-1$$ (we choose this formula since there is already a number $1$ before $\cos2x$, so using this formula can get rid of number $1$)
$$\sin2x=2\sin x\cos x$$
Replace $\cos2x$ and $\sin2x$ in $X$ with these, we have
$$X=\frac{1+2\cos^2x-1}{2\sin x\cos x}$$
$$X=\frac{2\cos^2x}{2\sin x\cos x}$$
$$X=\frac{\cos x}{\sin x}$$
Finally, using the Quotient Identity: $\frac{\cos x}{\sin x}=\cot x$, we have
$$X=\cot x$$
Hence, $$\frac{1+\cos2x}{\sin2x}=\cot x$$
The equation is an identity, because the two sides are equal.
