Question

Prove that sin squarex+cos square x can never be less than 2

Answer 1

as you know,
sin²x , cosec²x both are always positive numbers .
but we also know,
for any positive real numbers,
AM ≥ GM
where AM is arithmetic mean
and GM is Geometric mean

now,
AM of { sin²x, cosec²x} = (sin²x + cosec²x)/2
GM of { sin²x, cosec²x} = √(sin²x.cosex²x)
= √(sin²x/sin²x) = 1 [ because sinx = 1/cosecx ]

so,
(sin²x + cosec²x)/2 ≥ 1
sin²x +cosec²x ≥ 2
hence, proved /