Question

The equation $\sin \left(\cos x\right)=\cos \left(\sin x\right)$ has

• A. no roots in $[0,\frac{\pi }{2}]$
• B. no roots in $[\frac{\pi }{2},\pi ]$
• C. no roots in $[\pi ,\frac{3\pi }{2}]$
• D. no roots in $(-\infty ,\infty )$

Answer 1

The correct option is D no roots in $(-\infty ,\infty )$

For this to be true, $cosx=sinx=\frac{\pi }{4}$ and since there exists no such $x$ for which $sinx=cosx=\frac{\pi }{4}$, Thus no roots in $(-\infty ,\infty )$