${\cot\left({x}\right)}^{2} \times \left( {\sec\left({x}\right)}^{2}-1 \right)$
Use $\cot\left({t}\right)=\frac{ \cos\left({t}\right) }{ \sin\left({t}\right) }$ to transform the expression${\left( \frac{ \cos\left({x}\right) }{ \sin\left({x}\right) } \right)}^{2} \times \left( {\sec\left({x}\right)}^{2}-1 \right)$
Use $\sec\left({t}\right)=\frac{ 1 }{ \cos\left({t}\right) }$ to transform the expression${\left( \frac{ \cos\left({x}\right) }{ \sin\left({x}\right) } \right)}^{2} \times \left( {\left( \frac{ 1 }{ \cos\left({x}\right) } \right)}^{2}-1 \right)$
To raise a fraction to a power, raise the numerator and denominator to that power$\frac{ {\cos\left({x}\right)}^{2} }{ {\sin\left({x}\right)}^{2} } \times \left( {\left( \frac{ 1 }{ \cos\left({x}\right) } \right)}^{2}-1 \right)$
To raise a fraction to a power, raise the numerator and denominator to that power$\frac{ {\cos\left({x}\right)}^{2} }{ {\sin\left({x}\right)}^{2} } \times \left( \frac{ 1 }{ {\cos\left({x}\right)}^{2} }-1 \right)$
Write all numerators above the common denominator$\frac{ {\cos\left({x}\right)}^{2} }{ {\sin\left({x}\right)}^{2} } \times \frac{ 1-{\cos\left({x}\right)}^{2} }{ {\cos\left({x}\right)}^{2} }$
Use $1-{\cos\left({t}\right)}^{2}={\sin\left({t}\right)}^{2}$ to simplify the expression$\frac{ {\cos\left({x}\right)}^{2} }{ {\sin\left({x}\right)}^{2} } \times \frac{ {\sin\left({x}\right)}^{2} }{ {\cos\left({x}\right)}^{2} }$
Cancel out the common factor ${\cos\left({x}\right)}^{2}$$\frac{ 1 }{ {\sin\left({x}\right)}^{2} } \times {\sin\left({x}\right)}^{2}$
Cancel out the common factor ${\sin\left({x}\right)}^{2}$$1$
Since the expression is equal to the initial right-hand side, the identity is verified$\textnormal{True}$