Question

If $w={\log }_{\sin 1}\cos 1,x={\log }_{\sin 1}\tan 1,y={\log }_{\cos 1}$ $\sin 1$ and $z={\log }_{\cos }\tan 1$ then

• A. $w<y<x<z$
• B. $y<z<w<z$
• C. $x<z<y<w$
• D. $z<x<w<y$

Answer 1

Answer: (C)

Since $\frac{\pi }{4}<1,\cos 1<\sin 1<1<\tan 1$
Hence ${\log }_{\sin 1}\tan 1<{\log }_{\cos 1}\tan 1<0$
$0<{\log }_{\cos 1}\sin 1<{\log }_{\cos 1}\cos 1=1$
and $1={\log }_{\sin 1}\sin 1<{\log }_{\sin 1}\cos 1$
$\therefore x<z<y<w$