# Solve for: (tan(β)+cot(β))^2-csc(β)^2=sec(β)^2

## Expression: ${\left( \tan\left({β}\right)+\cot\left({β}\right) \right)}^{2}-{\csc\left({β}\right)}^{2}={\sec\left({β}\right)}^{2}$

Start working on the left-hand side

${\left( \tan\left({β}\right)+\cot\left({β}\right) \right)}^{2}-{\csc\left({β}\right)}^{2}$

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

$\textnormal{True}$

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