Question

least value of 25 cosec^2 X + sec^2 x​

Answer 1

Answer:

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Step-by-step explanation:

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Answer 2

Step-by-step explanation:

Let f(x) = cosec²x + 25sec²x

we know,

sec²x - tan²x = 1 , sec²x = 1 + tan²x

cosec²x - cot²x = 1 , cosec²x = 1 + cot²x

then, f(x) = 1 + cot²x + 25(1 + tan²x)

= 1 + cot²x + 25 + 25tan²x

f(x) = 26 + cot²x + 25tan²x

we know, if a and b are two positive terms,

then, AM ≥ GM

here cot²x and 25tan²x are two positive terms,

so, (cot²x + 25tan²x)/2 ≥ √{25tan²x.cot²x}

cot²x + 25tan²x ≥ 2 × √{25 × 1} = 2×5 = 10

cot²x + 25tan²x ≥ 10

add 26 both sides,

cot²x + 25tan²x + 26 ≥ 10 + 26 = 36

f(x) = cot²x + 25tan²x + 26 ≥ 36

hence, minimum value of f(x) is 36 .

hence, minimum value of cosec²x + 25sec²x = 36