First slide

Trigonometric Identities

Question

The greatest among $(\sin 1 + \cos 1), (\sqrt{\sin 1} + \sqrt{\cos 1}), (\sin 1 - \cos 1)$ and 1 is

Moderate

A

$\sin 1 + \cos 1$
B

$\sqrt{\sin 1} + \sqrt{\cos 1}$
C

$\sin 1 - \cos 1$
D

1

Solution

$\sqrt{\sin 1} > \sin 1 > \sin^{2} 1, \sqrt{\cos 1} > \cos 1 > \cos^{2} 1$
Adding $(\sqrt{\sin 1} + \sqrt{\cos 1}) > (\sin 1 + \cos 1) > (\sin^{2} 1 + \cos^{2} 1)$
$\Rightarrow (\sqrt{\sin 1} + \sqrt{\cos 1}) > (\sin 1 + \cos 1) > 1$

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