prove this: - - QuestionCove
OpenStudy (maheshmeghwal9):

prove this: -

4 years ago
OpenStudy (maheshmeghwal9):

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

4 years ago
OpenStudy (maheshmeghwal9):

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

4 years ago
OpenStudy (maheshmeghwal9):

now how to prove? Plz help :)

4 years ago
hartnn (hartnn):

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

4 years ago
OpenStudy (maheshmeghwal9):

ya

4 years ago
hartnn (hartnn):

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

4 years ago
hartnn (hartnn):

i will write 'x' instead of theta for simplicity

4 years ago
OpenStudy (maheshmeghwal9):

$-2 \le \sin x \cos x \le 2$

4 years ago
hartnn (hartnn):

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

4 years ago
OpenStudy (maheshmeghwal9):

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

4 years ago
hartnn (hartnn):

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

4 years ago
hartnn (hartnn):

**square all sides ** doing reciprocal :P

4 years ago
hartnn (hartnn):

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

4 years ago
OpenStudy (maheshmeghwal9):

why ?

4 years ago
hartnn (hartnn):

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

4 years ago
OpenStudy (maheshmeghwal9):

ya ?

4 years ago
OpenStudy (turingtest):

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

4 years ago
OpenStudy (maheshmeghwal9):

its ok :D

4 years ago
hartnn (hartnn):

ok, what you got after simplifying Left side ?

4 years ago
OpenStudy (maheshmeghwal9):

after opening the brackets ?

4 years ago
hartnn (hartnn):

in terms of cosec ,sec and constant

4 years ago
OpenStudy (maheshmeghwal9):

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

4 years ago
hartnn (hartnn):

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

4 years ago
hartnn (hartnn):

you can post few steps at a time...

4 years ago
OpenStudy (maheshmeghwal9):

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$

4 years ago
hartnn (hartnn):

ok, great!

4 years ago
hartnn (hartnn):

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 ??

4 years ago
hartnn (hartnn):

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

4 years ago
OpenStudy (maheshmeghwal9):

ok

4 years ago
hartnn (hartnn):

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

4 years ago
OpenStudy (maheshmeghwal9):

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

4 years ago
hartnn (hartnn):

which is 5+... ?

4 years ago
OpenStudy (maheshmeghwal9):

$5+\ge4$

4 years ago
hartnn (hartnn):

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

4 years ago
OpenStudy (maheshmeghwal9):

ok i gt it thanx:)

4 years ago
hartnn (hartnn):

welcome ^_^

4 years ago