$\tan\left({y}\right)\cot\left({x}\right)-1$
Use $\tan\left({t}\right)=\frac{ \sin\left({t}\right) }{ \cos\left({t}\right) }$ to transform the expression$\frac{ \sin\left({y}\right) }{ \cos\left({y}\right) } \times \cot\left({x}\right)-1$
Use $\cot\left({t}\right)=\frac{ \cos\left({t}\right) }{ \sin\left({t}\right) }$ to transform the expression$\frac{ \sin\left({y}\right) }{ \cos\left({y}\right) } \times \frac{ \cos\left({x}\right) }{ \sin\left({x}\right) }-1$
To multiply the fractions, multiply the numerators and denominators separately$\frac{ \sin\left({y}\right)\cos\left({x}\right) }{ \cos\left({y}\right)\sin\left({x}\right) }-1$
Write all numerators above the common denominator$\frac{ \sin\left({y}\right)\cos\left({x}\right)-\cos\left({y}\right)\sin\left({x}\right) }{ \cos\left({y}\right)\sin\left({x}\right) }$
Use $\sin\left({t}\right)\cos\left({s}\right)-\cos\left({t}\right)\sin\left({s}\right)=\sin\left({t-s}\right)$ to simplify the expression$\frac{ \sin\left({y-x}\right) }{ \cos\left({y}\right)\sin\left({x}\right) }$
Since the expression is equal to the initial left-hand side, the identity is verified$\textnormal{True}$