Prove-that-for-any-acute-angle-sec-cosec-tan-cot-4-

Question Number 8105 by 314159 last updated on 30/Sep/16

P r o v e t h a t, f o r a n y a c u t e a n g l e α

s e c α c o s e c α + t a n α + c o t α ⩾ 4 .

Commented by Yozzia last updated on 30/Sep/16

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

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

f (x) = \frac{2}{c o s s i n x} = \frac{4}{s i n 2 x} = 4 c o s e c 2 x

S i n c e 0 < α < 90 ° \Rightarrow 0 < 2 α < 180 °

\Rightarrow 0 < s i n 2 x ⩽ 1 \Rightarrow c o s e c 2 x ⩾ 1

\Rightarrow 4 c o s e c 2 x ⩾ 4 \Rightarrow f (x) ⩾ 4

∴ s e c x c o s e c x + t a n x + c o t x ⩾ 4 \forall x \in (0, 90 °) .

Answered by prakash jain last updated on 02/Oct/16

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