Question

the least value of 8 cosec²x + 25 sin²x ​

Answer 1

Answer:

1 is the answer of this questions

Answer 2

$\large\underline{\sf{To\:Find - }}$

The least value of 8 cosec²x + 25 sin²x

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\: {8cosec}^{2}x + {25sin}^{2}x$

We know,

$\boxed{ \bf{ \: cosecx \: = \: \frac{1}{sinx}}}$

So, above expression can be rewritten as

$\rm \:  =  \:  \: \dfrac{8}{ {sin}^{2} x} + 25 {sin}^{2}x$

We know,

If x and y are two positive real numbers, then

Arithmetic Mean between them is greater than or equals to Geometric Mean. i.e.

$\boxed{ \bf{ \: \dfrac{x + y}{2} \geqslant \sqrt{xy}}}$

or

$\boxed{ \bf{ \: x + y \geqslant 2 \sqrt{xy}}}$

So, using this identity, we get

$\rm \: \: \geqslant \: 2 \: \sqrt{\dfrac{8}{ {sin}^{2} x} \times {25sin}^{2}x}$

$\rm \: \: \geqslant \: 2 \: \sqrt{8 \times 25}$

$\rm \: \: \geqslant \: 2 \: \sqrt{200}$

$\rm \: \: \geqslant \: 2 \: \sqrt{2 \times 10 \times 10}$

$\rm \: \: \geqslant \: 20 \: \sqrt{2}$

Hence,

$\boxed{ \bf{ \: \: Least \: value \: of \: {8cosec}^{2}x + {25sin}^{2}x = 20 \sqrt{2}}}$

Additional Information :

If x and y are two positive real numbers then

$\boxed{ \bf{ \: AM = \frac{x + y}{2}}}$

$\boxed{ \bf{ \: GM = \sqrt{xy}}}$

$\boxed{ \bf{ \: HM = \frac{2xy}{x + y}}}$

and

$\boxed{ \bf{ \: \: AM \: \geqslant \: GM \: \geqslant \: HM \: }}$