Question

If $e^{-\frac{\pi }{2}}<\theta <\pi /2$, then-

• A. $\cos \log \theta <\log \cos \theta$
• B. $\cos \log \theta >\log \cos \theta$
• C. $\cos \log \theta \le \log \cos \theta$
• D. $\cos \theta >\log \theta$

Answer 1

The correct option is B $\cos \log \theta >\log \cos \theta$

$e^{-\frac{\pi }{2}}<\theta <\frac{\pi }{2}$

$\ln e^{-\frac{\pi }{2}}<\ln \theta <\ln \frac{\pi }{2}$

$\frac{-\pi }{2}<\ln \theta <\ln \frac{\pi }{2}$

$\cos \left(\frac{-\pi }{2}\right)<\cos \left(\ln \theta \right)<\cos \left(\ln \frac{\pi }{2}\right)$

$\therefore \cos \left(\ln \theta \right)>0$

$-1\le \cos \theta \le 1$

$\ln (-1)\le \ln \left(\cos \theta \right)\le \ln \left(1\right)$

$\therefore \ln \left(\cos \theta \right)\le 0$

$\cos \left(\ln \theta \right)>\ln \left(\cos \theta \right)\Rightarrow \cos \left(\log \theta \right)>\log \left(\cos \theta \right)$