First slide

Extreme values of trigonometric functions in trigonometry

Question

If $y = (\sin x + cosec x)^{2} + (\cos x + \sec x)^{2}$ then the minimum value of y, $\forall x \in R$ is

Moderate

A

7
B

3
C

9
D

0

Solution

$y = (\sin^{2} x + \cos^{2} x) + 2 (\sin xcosec x + \cos xsec x) + \sec^{2} x + {cosec}^{2} x$
$\begin{array}{l} = 5 + 2 + \tan^{2} x + \cot^{2} x \\ = 7 + (\tan x - \cot x)^{2} + 2 \\ ∴ y_{min} = 9 \end{array}$

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