Question

20.
(sin x+ cosec x)"2+(cos x+ secx)"2? is
(a) >_9
(b) <_9
(c) = 9
(d) none of these​

Answer 1

Answer:

(sinx + cosecx)^2 + (cosx + secx)^2

sin^2x + 2sinxcosex + cosec^2x + cos^2x + 2cosxsecx + sec^2x

sin^2x + cos^2x + 2 + 2 + cosec^2x + sec^2x

1 + 2 + 2 + cosec^2x + sec^2x

5 + cosec^2x + sec^2x

now, minimum value of cosecx is 1 when x is 90°. But at this value of x, value of secx is maximum i.e. infinity. Similarly, minimum value of secx is 1 when x is 0° but at this value of x, value of cosecx is maximum i.e. infinity.

Hence in either case, value of cosec^2x + sec^2x will be infinity.

The value of cosec^2x + sec^2x will be minimum when cosecx = secx for some x, and that value of x is 45° (we can derive this using maxima and minima)

cosec 45° = sec 45° = √2

so minimum value of expression is

5 + (√2)^2 + (√2)^2

5 + 2 + 2

9

hence, the value of expression will be equal to or more than 9 (option a).