I need to solve cos2x - tanx = 1, and it's driving me crazy.?

Question

i know the actual answers, but need the step by step process. helpz! thanks!

Answer 1

cos2x - tanx = 1

cos2x - 1 = tanx

1 - 2sin^2 x - 1 = tanx..................since cos2x = 1 - 2sin^2 x

- 2sin^2 x = sinx/cosx

- 2 sinx = 1/cosx

2sinx.cosx = -1

sin2x = -1

and solve further

that is

x = (sin inverse of -1)/2

x = 270/2

x = 135 degree

Answer 2

cos2x - tanx = 1
=>cos2x - 1 = tanx
=> -2(sinx)^2 = tanx
=> tanx + 2(sinx)^2 = 0
=> sinx / cosx + 2(sinx)^2 = 0
=> sinx(1/cosx + 2sinx) = 0
=> Either sinx=0, or, (1/cosx + 2sinx) = 0
=> Either x = n*pi, or, (1/cosx + 2sinx) = 0, where n is any integer.

The solution of the second part is:-
(1+2sinx cosx) / cosx=0
=> 1 + 2sinx(1-sin^2 x)^1/2 = 0
=> 2sinx(1-sin^2 x)^1/2 = -1
=> 4sin^2 x(1 - sin^2 x) = 1 [Squaring both sides]
Let sin^2 x = y
Then, 4y(1-y) = 1
=> 4y - 4y^2 = 1
=> 4y^2 - 4y + 1 = 0
=> (2y - 1)^2 = 0
=> 2y - 1 = 0
=> y=1/2
=> sin^2 x = 1/2
=> sin x = +or- 1/2
=> x = +or- (n*pi + pi/6), where n is any integer.

Hence, the three general solutions to your equation are x=n*pi, x=n*pi + pi/6, and, x=-(n*pi + pi/6)

Note:- It may be the case that you don't know what a general solution is right now. In that case, the answers are x=0, x=30 degrees, and x=-30 degrees. It is then sufficient for you to know that pi (in radians)= 180 degrees.

Answer 3

cos2x-tanx=1
cos2x=1+tanx
Just apply the formula and try now

Answer 4

cos2x-tanx=1

cos2x=cos(x+x)=cosx*cosx-sinx*sinx=(cosx-sinx)(cosx+sinx)
1+tanx=1+(sinx/cosx)=(cosx+sinx)/cosx

(cosx-sinx)(cosx+sinx)=1+tanx

(cosx-sinx)(cosx+sinx)=(cosx+sinx)/cosx

(cosx-sinx-1/cosx)(cosx+sinx)=0

CASE 1:
cosx-sinx-1/cosx=0
cos^2x-sinx*cosx-1=0 ( cosx is not 0 )
cos^2x-sinx*cosx-sin^2x-cos^2x=0
sinx(cosx+sinx)=0

(1) sinx=0
x=180*n (n is an integer)

(2) cosx+sinx=0
cos^2x=sin^2x
cos^2x=1-cos^2x
cos^2x=1/2
cosx=1/sqrt(2) or cosx=-1/sqrt(2)
(We must abandon first one.You will recognize the reason)
cosx=-1/sqrt(2)
x=-pi/4 + 2n*pi
x=(2n-1/4)*pi

CASE 2:
SAME AS SECOND SOLUTION OF CASE 1.

RESULT:
x=n*pi
x=(2n-1/4)*pi
(Here n is an integer)

Answer 5

x=n(22/7)

Answer 6

use this property
cos2x=(1-tan^2x)/(1+tan^2x)

=(1+tanx)(1-tanx)/(1+tan^2x)
now solve:
cos2x=1+tanx

Answer 7

don't go crazy for silly questions...
Leave them as they are and enjoy life