Let $PQRS$ be a quadrilateral in a plane, where $QR=1,\angle PQR=\angle QRS=70°,\angle PQS=15°$ and $\angle PRS=40°$. If $\angle RPS=\theta °,PQ=\alpha$ and $PS=\beta$, then the interval(s) that contain(s) the value of $4\alpha \beta \sin \theta °$ is/are

Given,

$PQRS$ be a quadrilateral in a plane, where $QR=1,\angle PQR=\angle QRS=70°,\angle PQS=15°$ and $\angle PRS=40°$. If $\angle RPS=\theta °,PQ=\alpha$ and $PS=\beta$, 

Now plotting the diagram of given values we get,

Applying sine rule in $∆PQR$ we get,

$\frac{\alpha }{\sin 30°}=\frac{1}{\sin 80°}$

$\Rightarrow \alpha =\frac{1}{2\sin 80°} ...\left(\mathrm{i}\right)$

Applying sine rule in $∆PRS$,

$\frac{\beta }{\sin {40}^2}=\frac{1}{\sin \theta }$

$\Rightarrow \beta \sin \theta =\sin 40° ...\left(\mathrm{ii}\right)$

From (i) & (ii),

$4\alpha \beta \sin \theta =4·\frac{1}{2\sin 80°}·\sin 40°=\frac{1}{\cos 40°}$

Using $\cos 0°=1,\cos 30°=\frac{\sqrt{3}}{2},\cos 45°=\frac{1}{\sqrt{2}}$ and $\cos 60°=\frac{1}{2}$

Only Options (A) and (B) are correct.