Solve the following equation
tan (2x)+ sec (2x) = cos (x)+ sin (x)
Thanks for your help
your identity is false
just take x=30º
LS = tan60 + 1/cos60
=sqrt(3) + 2
RS = cos30 + sin30
= sqrt(3)/2 + 1/2 which is not= to the LS
To show an identity to be false, all you need is one exception.
I am asked to solve this equation, not to prove this identities.
blame it on the fact I did not have my second cup of coffee yet.
sin2x/cos2 + 1/cos2x = sinx + cosx
(2sinxcosx + 1)cos2x = sinx + cosx
(2sinxcos + sin^2x + cos^2x)/(cos^2x-sin^2) = sinx + cosx
(sinx+cosx)^2/(cos^2-sin^2) = sinx + cosx
(sinx + cosx)/[cosx-sinx)(cosx+sinx)] = sinx + cosx
crossmultiply
(sinx + cosx)^2 = (cosx+sinx)^2(cosx-sinx)
divide both sides by (cosx+sinx)^2
1 = cosx-sinx
sinx = cosx
divide by cosx
tanx = 1
x = 45º or 225º or pi/4, 5pi/4
Ahhh, but in the original that would make it tan 90º which of course is undefined.
SO THERE IS NO SOLUTION TO YOUR EQUATION
There is at least one solution: x = 0
tan (0)+ sec (0) = cos (0)+ sin (0)
0 + 1 = 1 + 0
Any multiple of 360 degrees will also work.
I've check my answer key and the answers are 0, 270 and 360. I cant figure out how to do it.
btw , are drwls and reiny teachers or professors ??
Yes. One is a retired teacher; the other is a retired professor.
Actually I am a retired PhD physicist/engineer. I worked in the aerospace industry until 1992, and have been doing online tutoring since 1994.
To solve the equation: tan(2x) + sec(2x) = cos(x) + sin(x), we will simplify it using trigonometric identities and solve for x.
Step 1: Rearrange the equation
tan(2x) + sec(2x) - (cos(x) + sin(x)) = 0
Step 2: Replace tan(2x) and sec(2x) using trigonometric identities
sin(2x)/cos(2x) + 1/cos(2x) - (cos(x) + sin(x)) = 0
Step 3: Combine the terms on the left side of the equation
(sin(2x) + 1 - cos(2x)cos(x) - cos(2x)sin(x))/cos(2x) = 0
Step 4: Simplify and expand
(2sin(x)cos(x) + 1 - (1 - 2sin^2(x))) / (1 - 2sin^2(x)) = 0
(2sin(x)cos(x) + 2sin^2(x)) / (1 - 2sin^2(x)) = 0
Step 5: Factor out sin(x)
sin(x)(2cos(x) + 2sin(x)) / (1 - 2sin^2(x)) = 0
Step 6: Set each term equal to zero
sin(x) = 0 (1)
2sin(x) + 2cos(x) = 0 (2)
Step 7: Solve equation (1)
sin(x) = 0
The solutions for sin(x) = 0 are x = 0, π, -π, 2π, etc.
Step 8: Solve equation (2)
2sin(x) + 2cos(x) = 0
Divide by 2: sin(x) + cos(x) = 0
Apply the Pythagorean identity: sin(x) + cos(x) = √2 sin(x + π/4) = 0
Set sin(x + π/4) = 0: x + π/4 = nπ where n is an integer
Solve for x: x = nπ - π/4 where n is an integer
So, the solutions for sin(x) + cos(x) = 0 are x = nπ - π/4 where n is an integer.
The final solution for the equation tan(2x) + sec(2x) = cos(x) + sin(x) is the combination of the solutions obtained in steps 7 and 8.