How do you solve #cscx+cotx=1# and find all solutions in the interval #[0,2pi)#?

Question

Josiah Anderson · Accepted Answer

#1/sinx + cosx/sinx =1# #(1 + cosx)/sinx = 1/1# #1+ cosx = sinx# #(1 + cosx)^2 = (sinx)^2# #1 + 2cosx + cos^2x = sin^2x# #1 + 2cosx + cos^2x = 1 - cos^2x# #2cos^2x + 2cosx + 1 - 1 = 0# #2cosx(cosx + 1) = 0# #cosx = 0 and cosx = -1# #x= pi/2, (3pi)/2, pi# However, checking in the original equation, you will find that #x = (3pi)/2# is extraneous and #x = pi# makes the equation undefined and is thus also extraneous.  Hence, the only solution in the interval #[0, 2pi)# is #{pi/2}#. Hopefully this helps!

Nolan Alexander · Answer

#color(red)("solution is "x=pi/2)#

Given equation

#cscx+cotx=1....(1)#

Now we know

#csc^2x-cot^2x=1#

#=>(cscx+cotx)(cscx-cotx)=1#

#=>1*(cscx-cotx)=1#

#=>(cscx-cotx)=1....(2)#

Adding (1) & (2) we get

#2cscx=2=>cscx=1=csc(pi/2)#

#:.x=pi/2#

Again Subtracting (2) from (1)

#2cotx=0=>cotx=0=cot(pi/2)# #:.x=pi/2#

For #cotx=0=cot(3pi/2)# then #x=3pi/2# but this does not satisfy the given equation as #sin(3pi/2)=-1#

#color(red)("So only solution is "x=pi/2)#

Alternative

Given equation

#cscx+cotx=1#

#=>1/sinx+cosx/sinx=1#

#=>(1+cosx)/sinx=1#

#=>sinx-cosx=1#

#=>1/sqrt2*sinx-1/sqrt2*cosx=1/sqrt2#

#=>sin(pi/4)*sinx-sin(pi/4)*cosx=1/sqrt2#

#=>sin(x-pi/4)=sin(pi/4)#

#=>x=pi/4+pi/4=pi/2#

Again

#=>sin(x-pi/4)=1/sqrt2=sin(3pi/4)#

#=>x=(3pi)/4+pi/4=pi# But this does not satisfy the given equation as #cscpi and cot pi" undefined"#

#color(red)("So only solution is "x=pi/2)#

Grace Ashcraft · Answer

undefined To solve the equation csc(x) + cot(x) = 1 and find all solutions in the interval [0, 2π), follow these steps:

1. Rewrite csc(x) and cot(x) in terms of sine and cosine:
   csc(x) = 1/sin(x)
   cot(x) = cos(x)/sin(x)

2. Substitute these expressions into the equation:
   1/sin(x) + cos(x)/sin(x) = 1

3. Combine the fractions on the left side of the equation:
   (1 + cos(x))/sin(x) = 1

4. Cross multiply to eliminate the fraction:
   1 + cos(x) = sin(x)

5. Rearrange the equation:
   sin(x) - cos(x) - 1 = 0

6. Use trigonometric identities to rewrite sin(x) and cos(x):
   sin(x) = sqrt(1 - cos^2(x))
   cos(x) = sqrt(1 - sin^2(x))

7. Substitute these expressions into the equation:
   sqrt(1 - cos^2(x)) - cos(x) - 1 = 0

8. Square both sides to eliminate the square roots:
   1 - cos^2(x) - 2cos(x) + cos^2(x) - 1 = 0

9. Simplify the equation:
   -2cos(x) = 0

10. Solve for cos(x):
   cos(x) = 0

11. Find the solutions for x in the interval [0, 2π) where cos(x) = 0:
   x = π/2, 3π/2

Therefore, the solutions to the equation csc(x) + cot(x) = 1 in the interval [0, 2π) are x = π/2 and x = 3π/2.