Question

prove this: -

Answer 1

\[(\sin \theta + cosec \theta)^2 +(\cos \theta + \sec \theta)^2 \ge 9\]

Answer 2

I did only this much
\[(\sin \theta + cosec \theta)^2 +(\cos \theta + \sec \theta)^2=\tan^2 \theta + \cot^2 \theta +7.\]

Answer 3

now how to prove?
Plz help :)

Answer 4

ok, do you know that range of sin is between 1 and -1 ?

Answer 5

ya

Answer 6

so, start with
\( -1 \le \sin 2x \le 1\)
and apply the formula for sin 2x=... ?
what you get ?

Answer 7

i will write 'x' instead of theta for simplicity

Answer 8

\[-2 \le \sin x \cos x \le 2\]

Answer 9

no no....you multipled by 2 instead of dividing...

Answer 10

oops sorry \[-\frac{1}{2} \le \sin x \cos x \le \frac{1} {2}\]

Answer 11

now do these 2 things,
1) square both sides,
2) take the reciprocal
don't forget to flip the sign while reciprocalling..

Answer 12

**square all sides
** doing reciprocal :P

Answer 13

it will be
\(cosec^2 x \sec^2 x \ge 4 \)
only.

Answer 14

why ?

Answer 15

how can that be greater than 4 and less than 4 at the same time ?

Answer 16

ya ?

Answer 17

oops, misread the question lol
carry on... :P

Answer 18

its ok :D

Answer 19

ok,
what you got after simplifying Left side ?

Answer 20

after opening the brackets ?

Answer 21

in terms of cosec ,sec and constant

Answer 22

after simplyfying; i gt
\[\tan ^2 \theta + \cot ^2 \theta +7\]

Answer 23

in terms of sec and cosec.....
ok, just open the brackets , what you get ?

Answer 24

you can post few steps at a time...

Answer 25

i gt this after opening up the brackets
\[\sin^2 \theta + \cos^2 \theta + 2\sin \theta cosec \theta + 2 \cos \theta \sec \theta +\sec^2 \theta + cosec ^2 \theta\]\[1 + 2 + 2 + \sec ^2 \theta + cosec^2 \theta\]\[5+ 1+\tan^2 \theta+1+\cot^2 \theta\]\[7+\tan ^2 \theta + \cot^2 \theta\]

Answer 26

ok, great!

Answer 27

first read this :
i will get to \(cosec^2 x \sec^2 x \ge 4\) by another understandable method.

\(-1 \le \sin 2x \le 1 \\squaring, 0 \le sin^2 2x \le 1\)
because square of a number can't be negative.
\(\implies 0 \le sin^2 x \cos^2 x \le 1/4 \)
now taking the reciprocals,
\(4\le cosec^2 x \sec^2 x \le \infty \implies cosec^2 x \sec^2 x \ge 4\)
did you get this ??

Answer 28

and i wanted you to stop at
\(1 + 2 + 2 + \sec ^2 \theta + cosec^2 \theta \)

Answer 29

ok

Answer 30

now write sec^2 = 1/ cos^2 and cosec^2 = 1/sin^2
what u get ?

Answer 31

\[5+ \frac{1}{\sin^2 \theta \cos^2\theta}\]

Answer 32

which is 5+... ?

Answer 33

\[5+\ge4\]

Answer 34

Make sure you got this first,
\(4\le cosec^2 x \sec^2 x \le \infty \implies cosec^2 x \sec^2 x \ge 4 \\ \implies 5+ cosec^2 x \sec^2 x \ge9 \)

Answer 35

ok i gt it thanx:)

Answer 36

welcome ^_^