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Trigonometrical Equations
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Minimum value of the function $f(x) = \left| {\sin \,x + \cos \,x + \tan \,x + \cot \,x + \sec \,x + \ cosec\ x} \right|$ is equal to
A
$2\sqrt 2$
B
$2\sqrt 2 - 1$
C
$2 + 3\sqrt 2 $
D
$2\sqrt 2 + 1$
Solution
Put $=\sin x+\cos x=p \Rightarrow 1+2 \sin x \cos x=p^{2}$
$\sin x \cos x=\left(\frac{p^{2}-1}{2}\right)$
$\therefore f(x) = \left| {p + \frac{2}{{\left( {{p^2} – 1} \right)}} + \frac{{2p}}{{{p^2} – 1}}} \right|$
${=\left|p+\frac{2}{(p-1)}\right|} $
${=\left|(p-1)+\frac{2}{(p-1)}+1\right|}$
$\therefore \frac{(p-1)+\left(\frac{2}{p-1}\right)}{2} \geq\left((p-1) \cdot \frac{2}{p-1}\right)^{1 / 2}$
$\therefore(p-1)+\left(\frac{2}{p-1}\right) \geq 2 \sqrt{2}$
$\therefore \quad f(x)_{\min } \geq 2 \sqrt{2}+1$
Standard 11
Mathematics
