Question

if sinx+cosx=sqrt(cosx) then prove that cosx-sinx=sqrt(sinx), this is from cbse class 11 problem

Answer 1

Try squaring both sides first.

Answer 2

(hint: this will only be true for certain values of x..)

Answer 3

thanks i tried squaring on both sides but no help

Answer 4

=>multiply both sides by 1/sqrt(2)

=>sin(x + pi/4) = sqrt(cos x /2)
=> sin^2 (x +pi/4) = cosx /2
=>2 sin^2 (x + pi/4) = cos x
=> 1- 2 sin^2(x+ pi/4) = 1 - cos x
=> cos ( 2x + pi/2) = 1- cosx
=> - cos( 2x) = 1- cos x
=> -2 cos^2 x +1 = 1 - cos x

=>cos^2 x = cos x
=>cos x = 0 , 1
both values dont satisfy original eqn,,
where did i go wrong? o.O

Answer 5

wel,,cos x =1 satisfy original eqn, but not the latter one..
you sure the ques is right ? or please correct me in case i went wrong ?

Answer 6

this question is correct only, it was given in the recent test for class 11 and i also have one more similar question that is:
if tan square x = sinx * cosx then prove 2 - sin to the power of 6 x = 4 * sin square x = 3 * sin to the power of 4 x =1

Answer 7

ohh..lol//
i will have
2 cos^2 x = cos x
=> cos x = 0
=>cos x = 1/2

when cos x =1/2, sin x = sqrt(3)/2 or - sqrt(3)/2

cos x = 0 isnt a solution,,

maybe latter one satisfies it..

please confirm anyone ?
and sorry for confusion above /

Answer 8

seems even latter one doesnt satisfy! :|

Answer 9

Well there is no solution of x......I think we just need to prove from the given condition

Answer 10

if no x satisfies both,,how can it be true ?

Answer 11

sinx + cosx = sqr(cosx)
1+2sin2x = cosx .....i
cosx -sinx =sqr(sinx)
1-2sin2x = sinx....ii
Now, from i and ii
cosx + sinx =2 .......(which has no solution)

Answer 12

aha..cool..

conclusion will be given statement is false right ?

Answer 13

but, (sinx+cosx)^2 = 1+sin2x not 1+2sin2x

Answer 14

@RadEn true !!

but it wont effect the final eqn..

Answer 15

sinx + cosx = sqrt(cosx)
1+sin2x = cosx
sin2x = cosx - 1 ... i
let k=cosx-sinx
k^2 = 1-sin2x .... ii
subs (i) to (ii)
k^2 = 1-(cosx - 1)
k^2 = 2-cosx
k=sqrt(2-cosx)
not prove! lol

Answer 16

just a try can u use (a+b)^2 = (a-b)^2+4ab after squaring on both sides ...

Answer 17

Not getting...

Answer 18

@hartnn , And I am not typing...

Answer 19

(sinx+cosx)^2 = cosx
(cosx=sinx)^2 + 4sinx cosx = cosx
(cosx=sinx)^2 = cosx - 2 2 sinx cosx
(cosx=sinx)^2 = cosx -2 sin2x
i dont know to reduce after this can anyone help.....

Answer 20

\[\cos^2(x) - \sin^2(x) = \cos(2x)\]

\[\implies (\cos(x) - \sin(x))(\cos(x) + \sin(x)) = \cos(2x)\]

\[\cos(x) - \sin(x) = \frac{\cos(2x)}{\sqrt{\cos(x)}} = ??\]

Answer 21

Prove RHS = \(\sqrt{sin(x)}\)