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  4. If y=( sin x+ textcosec x)2+( cos x+ sec x)2, then the minimum value of y, ∀ x ∈ R, is

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

Trigonometric Functions

Solution:

$y=\left(\sin ^{2} x+\cos ^{2} x\right)+2(\sin x\, \text{cosec} x+\cos x \sec x) +\sec ^{2} x+\text{cosec}^{2} x$
$=5+2+\tan ^{2} x+\cot ^{2} x$
$=7+(\tan x-\cot x)^{2}+2$
$\therefore y_{\min } =9$