# Solve for: cot(x)^2 * (sec(x)^2-1)=1

## Expression: ${\cot\left({x}\right)}^{2} \times \left( {\sec\left({x}\right)}^{2}-1 \right)=1$

Start working on the left-hand side

${\cot\left({x}\right)}^{2} \times \left( {\sec\left({x}\right)}^{2}-1 \right)$

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

$\textnormal{True}$

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