Question

If sin (π cos θ) = cos (π sin θ), then sin 2 θ =

(a) $\pm \frac{3}{4}$
(b) $\pm \frac{4}{3}$
(c) $\pm \frac{1}{3}$
(d) none of these

Answer 1

$$\begin{gathered} \left(\mathrm{c}\right)\pm \frac{1}{3} \\ \sin \left(\mathrm{\pi }\cos \mathrm{\theta }\right)=\cos \left(\mathrm{\pi }\sin \mathrm{\theta }\right) \\ \mathrm{As}\mathrm{we}\mathrm{know}\mathrm{that}\sin \theta =-\cos \left(\frac{\mathrm{\pi }}{2}+\theta \right) \\ \Rightarrow -\cos \left(\frac{\mathrm{\pi }}{2}+\mathrm{\pi }\cos \mathrm{\theta }\right)=\cos \left(\mathrm{\pi }\sin \mathrm{\theta }\right) \\ \Rightarrow \frac{-\mathrm{\pi }}{2}-\mathrm{\pi }\cos \mathrm{\theta }=\mathrm{\pi }\sin \mathrm{\theta } \\ \Rightarrow \mathrm{\pi }\sin \mathrm{\theta }-\mathrm{\pi }\cos \mathrm{\theta }=\frac{\mathrm{\pi }}{2} \\ \Rightarrow \sin \mathrm{\theta }-\cos \mathrm{\theta }=\frac{1}{2} \\ \mathrm{Squaring}\mathrm{both}\mathrm{sides}\mathrm{we}\mathrm{get}, \\ {\sin }^2\mathrm{\theta }+{\cos }^2\mathrm{\theta }-2\mathrm{sin\theta }\mathrm{cos\theta }=\frac{1}{4} \\ \Rightarrow 1-\sin 2\mathrm{\theta }=\frac{1}{4} \\ \Rightarrow \sin 2\mathrm{\theta }=\frac{1}{3} \\ \mathrm{And}\mathrm{we}\mathrm{know}\mathrm{that}\sin \mathrm{\theta }=\cos \left(\frac{\mathrm{\pi }}{2}-\mathrm{\theta }\right) \\ \Rightarrow \cos \left(\frac{\mathrm{\pi }}{2}-\mathrm{\pi }\cos \mathrm{\theta }\right)=\cos \left(\mathrm{\pi }\sin \mathrm{\theta }\right) \\ \Rightarrow \frac{\mathrm{\pi }}{2}-\mathrm{\pi }\cos \mathrm{\theta }=\mathrm{\pi }\sin \mathrm{\theta } \\ \Rightarrow \mathrm{\pi }\sin \mathrm{\theta }+\mathrm{\pi }\cos \mathrm{\theta }=\frac{\mathrm{\pi }}{2} \\ \Rightarrow \sin \mathrm{\theta }+\cos \mathrm{\theta }=\frac{1}{2} \\ \mathrm{Squaring}\mathrm{both}\mathrm{sides}\mathrm{we}\mathrm{get}, \\ \Rightarrow \sin {}^2\mathrm{\theta }+\cos {}^2\mathrm{\theta }+2\sin \mathrm{\theta }\cos \mathrm{\theta }=\frac{1}{4} \\ \Rightarrow 1+\sin 2\mathrm{\theta }=\frac{1}{4} \\ \Rightarrow \sin 2\mathrm{\theta }=\frac{1}{3} \\ \mathrm{Therefore},\sin 2\mathrm{\theta }=\pm \frac{1}{3} \end{gathered}$$