Question

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11. () Prove that tan x + cot x can never be equal
(u) Prove that sec2 x + cos2x can never be less than 2.​

Answer 1

Explanation:

11(ii) to solve this you need to remember a result obtained from AM-GM inequality (AM≥GM)

i.e,

if a>0,then a+1/a≥2 [always]

and if

a<0,

then a+1/a≤-2

proof of the above result:

AM≥GM

(a+1/a)/2≥√(a*1/a)

a+1/a≥2 [when a>0]

and

a+1/a≤-2 [when a<0]

so,

sec²x is positive

so,

sec²x+1/sec²x ≥ 2

sec²x + cos²x≥2 [SHOWN]

hence,sec²x +cos² x is never less than 2