Question

Solve $cosecx=1+\cot x$.

Answer 1

The given equation can be written as
$1/\left(sinx\right)=1+\left(\cos x/sinx\right)$,
provided $x\ne n\pi$ i.e., $\sin x\ne 0$
or $1=\sin x+\cos x$
or $\cos (x-\pi /4)=\cos \left(\pi /4\right)$
$\therefore x-\pi /4=2n\pi \pm \left(\pi /4\right)$
Hence $x=2n\pi$ or $x=2n\pi +\pi /2$, $n\in I$
But $x=2n\pi$ s ruled out
$[\because x\ne n\pi ]$
Hence we get the answer:
$x=2n\pi +\left(\pi /2\right),n\in I$.