Question

\cot ( x- \dfrac{ \pi }{ 2 } ) = 1

Answer 1

`\begin{array}{llll} \textit{Cofunction Identities} \\\\ \cot\left((\pi)/(2)-\theta\right)=\tan(\theta) \end{array}\hspace{5em} \begin{array}{llll} \textit{Symmetry Identities} \\\\ \cot(-\theta)=-\cot(\theta ) \end{array} \\\\[-0.35em] ~\dotfill`

`\cot\left( x-(\pi )/(2) \right)=1\implies -\cot\left( x-(\pi )/(2) \right)=-1\implies \cot\left[ -\left( x-(\pi )/(2) \right) \right]=-1 \\\\\\ \cot\left( (\pi )/(2)-x \right)=-1\implies \tan(x)=-1\implies x=\tan^(-1)(-1)\implies x= \begin{cases} (3\pi )/(4)\\\\ (7\pi )/(4) \end{cases}`