JEE Advance - Mathematics (1980 - No. 10)
For all $$\theta $$ in $$\left[ {0,\,\pi /2} \right]$$ show that, $$\cos \left( {\sin \theta } \right) \ge \,\sin \,\left( {\cos \theta } \right).$$
The inequality is false for all \(\theta \) in \([0, \pi/2]\).
The inequality holds because \(\cos(x)\) is always greater than \(\sin(x)\) for all \(x\).
The inequality is true because \(\cos(\sin \theta)\) and \(\sin(\cos \theta)\) are both bounded by 1.
The inequality holds because \(\cos(x) \ge \sin(x)\) for all \(x \in [0, \pi/4]\) and because of the properties of sine and cosine functions in the interval \([0, \pi/2]\).
The inequality holds only for \(\theta = 0\) and \(\theta = \pi/2\).