# Evaluate: (sin(y-x))/(cos(y)sin(x))=tan(y)cot(x)-1

## Expression: $\frac{ \sin\left({y-x}\right) }{ \cos\left({y}\right)\sin\left({x}\right) }=\tan\left({y}\right)\cot\left({x}\right)-1$

Start working on the right-hand side

$\tan\left({y}\right)\cot\left({x}\right)-1$

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

$\textnormal{True}$

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