First slide

Extreme values and Periodicity of Trigonometric functions

Question

For any real $θ$ the maximum value of $\cos^{2} (\cos θ) + \sin^{2} (\sin θ)$ is

Moderate

A

1
B

$1 + \sin^{2} 1$
C

$1 + \cos^{2} 1$
D

$2 + \cos^{2} 1$

Solution

$\cos^{2} (\cos θ) + \sin^{2} (\sin θ)$
$+ \sin^{2} (\cos θ) - \sin^{2} (\cos θ)$
$= 1 + \sin^{2} (s i n θ) - s i n^{2} (\cos θ),$ which is maximum when $\cos θ = 0, \sin θ = 1$

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