Answer
$45^{\circ}, 135^{\circ}$
Work Step by Step
Given $\sqrt 2 \csc\theta= 2$...(1)
we have to use the reciprocal identity that $\csc\theta $ in terms of $\sin\theta$
i.e. $\csc\theta = \frac{1}{\sin\theta}$...(2)
Substitute (2) in equation (1) we get
$\sqrt 2 \frac{1}{\sin\theta}= 2$
$\sqrt 2 = 2\sin\theta$
On dividing both sides by 2
$\frac{\sqrt 2}{2} = \frac{2\sin\theta}{2}$
$\frac{1}{\sqrt 2} = \sin\theta$
Therefore $\theta = 45^{\circ}$
sine function is positive in quadrant 1 and 3
So,
$\theta = 45^{\circ}$ or $\theta = 180^{\circ}-45^{\circ}$
$\theta = 45^{\circ}$ or $\theta = 135^{\circ}$
