Question

Find general solutions for 2tanx - cotx + 1 = 0

Answer 1

Check the answer in the following picture

Answer 2

Answer:

General Solution of the given equation is  $x=n\pi+tan^{-1}\,\frac{1}{2}$  and x = nπ + 3π/4

Step-by-step explanation:

Given Equation,

2 tan x - cot x + 1 = 0

To find: General Solution of the given equation.

Consider,

2 tan x - cot x + 1 = 0

2 tan x - 1/tan x + 1 = 0

2 tan² x - 1 + tan x = 0

2 tan² x + tan x - 1 = 0

2 tan² x + 2 tan x - tan x - 1 = 0

2 tan x ( tan x + 1 ) - ( tan x + 1 ) = 0

( 2tan x - 1 ) ( tan x + 1 ) = 0

⇒ 2 tanx - 1 = 0      and       tan x + 1 = 0

⇒ tan x = 1/2          and       tan x = -1

⇒ $x=tan^{-1}\,\frac{1}{2}$          and       tan x = tan 3π/4

⇒ $x=tan^{-1}\,\frac{1}{2}$          and       x = 3π/4

We know that when tan x = tan y

the general solution is given by , x = nπ + y

Here,

$x=n\pi+tan^{-1}\,\frac{1}{2}$  

and x = nπ + 3π/4

Therefore, General Solution of the given equation is  $x=n\pi+tan^{-1}\,\frac{1}{2}$  and x = nπ + 3π/4