Question

`tan((x/2)-(pi /2))=\sqrt2`

Answer 1

`\textit{Cofunction Identities} \\\\ sin\left(\theta-(\pi)/(2)\right)=-cos(\theta) \qquad\qquad cos\left(\theta-(\pi)/(2)\right)=+sin(\theta) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ tan\left((x)/(2)~~ - ~~(\pi )/(2) \right)~~ = ~~√(2)\qquad \qquad \qquad \stackrel{\textit{let's make for a second}}{\cfrac{x}{2}=\theta } \\\\[-0.35em] ~\dotfill`

`tan\left(\theta-(\pi)/(2)\right)\implies \cfrac{sin\left(\theta-(\pi)/(2)\right)}{cos\left(\theta-(\pi)/(2)\right)}\implies \cfrac{-cos(\theta )}{+sin(\theta )}\implies -cot(\theta )\implies -cot\left( (x)/(2) \right)`

`-cot\left( (x)/(2) \right)~~ = ~~√(2)\implies cot\left( (x)/(2) \right)=-√(2) \\\\\\ cot^(-1)\left[ cot\left( (x)/(2) \right) \right]=cot^(-1)\left(-√(2) \right)\implies \cfrac{x}{2}=cot^(-1)\left(-√(2) \right) \\\\[-0.35em] ~\dotfill\\\\ ~\hfill x=2\left[ cot^(-1)\left(-√(2) \right) \right]~\hfill`