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