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Extreme values and Periodicity of Trigonometric functions

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Question

If the minimum value of $∣ \sin x + \cos x + \tan x + \cot x + \sec x + cosec x ∣, x \in R$ is $\sqrt{m} + n$ then m + n equal $(m, n \in z)$

Difficult

Solution

Playing a little bit with the expression one discovers that if
we get $t = \sin x + \cos x, \frac{t^{2} - 1}{2} = sinx cosx$ ,then given expression $= |t + \frac{1}{\frac{t^{2} - 1}{2}} + \frac{t}{\frac{t^{2} - 1}{2}}|$
$\begin{array}{l} = |t + \frac{2}{t^{2} - 1} + \frac{2 t}{t^{2} - 1}| \\ y = |t + \frac{2}{t - 1}| = |t - 1 + \frac{2}{t - 1} + 1| \end{array}$
If $1 < t \leq \sqrt{2}$ then $y = t - 1 + \frac{2}{t - 1} + 1 \geq 2 \sqrt{2} + 1$
If $- \sqrt{2} \leq t < - 1$ then $y = - 1 + 1 - t + \frac{2}{1 - t} \geq 2 \sqrt{2} - 1$
$\begin{array}{l} ∴ y_{min} = 2 \sqrt{2} - 1 = \sqrt{8} - 1 \\ ∴ m + n = 7 \end{array}$

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