## Trigonometry (11th Edition) Clone

$$\frac{1+\cos2x}{\sin2x}=\cot x$$ The equation is an identity, because the two sides are equal.
$$\frac{1+\cos2x}{\sin2x}=\cot x$$ 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.