Minimum-value-of-sin-x-cos-x-tan-x-cot-x-sec-x-cosec-x-is-

Question Number 65192 by naka3546 last updated on 26/Jul/19

M i n i m u m v a l u e o f

∣ \sin x + \cos x + \tan x + \cot x + \sec x + cosec x ∣ i s \dots

Answered by Tanmay chaudhury last updated on 26/Jul/19

s i n x + c o s x \to (l e t s i n x + c o s x = a)

\sqrt{2} (\frac{1}{\sqrt{2}} s i n x + \frac{1}{\sqrt{2}} c o s x)

\sqrt{2} s i n (x + \frac{π}{4})

\sqrt{2} ⩾ \sqrt{2} s i n (x + \frac{π}{4}) ⩾ - \sqrt{2}

\sqrt{2} ⩾ a ⩾ - \sqrt{2}

s o

\sqrt{2} ⩾ s i n x + c o s x ⩾ - \sqrt{2}

t a n x + c o t x

= \frac{1}{s i n x c o s x} = \frac{2}{s i n 2 x} = \frac{2}{- 1 + 1 + s i n 2 x}

= \frac{2}{- 1 + {(s i n x + c o s x)}^{2}} = \frac{2}{a^{2} - 1}

s e c x + c o s e c x

= \frac{1}{c o s x} + \frac{1}{s i n x}

= \frac{2 (s i n x + c o s x)}{s i n 2 x}

= \frac{2 (a)}{a^{2} - 1}

n o w ∣ s i n x + c o s x + t a n x + c o t x + s e c x + c o s e c x ∣

=∣ a + \frac{2}{a^{2} - 1} + \frac{2 a}{a^{2} - 1} ∣

=∣ a + 2 \times \frac{a + 1}{(a + 1) (a - 1)} ∣

=∣ a + \frac{2}{a - 1} ∣

p u t a = \sqrt{2}

=∣ \sqrt{2} + \frac{2 (\sqrt{2} + 1)}{(\sqrt{2} - 1) (\sqrt{2} + 1)} ∣

=∣ \sqrt{2} + 2 \sqrt{2} + 2 ∣

= 3 \sqrt{2} + 2

= 6.23

p u t a = - \sqrt{2}

∣ - \sqrt{2} + \frac{2}{- \sqrt{2} - 1} ∣

∣ - \sqrt{2} - \frac{2 (\sqrt{2} - 1)}{1} ∣

∣ - \sqrt{2} - 2 \sqrt{2} + 2 ∣

∣ 2 - 3 \sqrt{2} ∣

∣ - 2.23 ∣

= 2.23

p u t a = 0

∣ a + \frac{2}{a - 1} ∣

=∣ 0 + \frac{2}{- 1} ∣

= 2

s o m i n i m u m v a l u e i s 2

p l s c h e c k