2tan x/1+tan2 x= sin(2x)
Starting with the left side of the equation, we have:
2tan x / (1 + tan^2 x)
Using the identity tan^2 x = 1 - cos^2 x, we can rewrite the denominator as:
2tan x / (1 + (1 - cos^2 x))
Simplifying further:
2tan x / (2 - cos^2 x)
Next, we can use the identity 1 - cos^2 x = sin^2 x to rewrite the denominator again:
2tan x / (2 - sin^2 x)
Now, let's focus on the right side of the equation:
sin(2x)
Using the double angle identity for sine, we have:
2sin x cos x
We can rewrite this as:
2sin x (1 - sin^2 x)
Now, equating the left and right sides of the equation:
2tan x / (2 - sin^2 x) = 2sin x (1 - sin^2 x)
Cross-multiplying:
2tan x (1 - sin^2 x) = 2sin x (2 - sin^2 x)
Expanding both sides:
2tan x - 2tan x sin^2 x = 4sin x - 2sin^3 x
Adding 2tan x sin^2 x to both sides:
2tan x = 4sin x - 2sin^3 x + 2tan x sin^2 x
Rearranging terms:
2tan x - 2tan x sin^2 x - 4sin x + 2sin^3 x = 0
Factoring out a common factor of 2sin x:
2sin x (tan x - tan x sin^2 x - 2 + sin^2 x) = 0
Taking out a common factor of sin x from the second term in parentheses:
2sin x (tan x - tan x sin^2 x - 2 + sin^2 x) = 0
Simplifying the second term:
2sin x (tan x - tan x sin^2 x - 2 + sin^2 x) = 0
2sin x (tan x - 3tan x sin^2 x - 2) = 0
Now, we can set each factor equal to zero:
sin x = 0 or (tan x - 3tan x sin^2 x - 2) = 0
The equation sin x = 0 has solutions x = 0, π, 2π, etc., which are the x-intercepts of the sine function.
To solve the second factor:
tan x - 3tan x sin^2 x - 2 = 0
We can factor out a common factor of tan x:
(tan x)(1 - 3sin^2 x) - 2 = 0
Now, we can use the identity sin^2 x = 1 - cos^2 x to substitute for sin^2 x:
(tan x)(1 - 3(1 - cos^2 x)) - 2 = 0
Simplifying:
(tan x)(1 - 3 + 3cos^2 x) - 2 = 0
(tan x)(-2 + 3cos^2 x) - 2 = 0
Expanding:
-2tan x + 3tan x cos^2 x - 2 = 0
Rearranging terms:
-2tan x + 3tan x cos^2 x = 2
Factoring out a common factor of tan x:
tan x (-2 + 3cos^2 x) = 2
Now, we can divide both sides by (-2 + 3cos^2 x):
tan x = 2 / (-2 + 3cos^2 x)
Therefore, the solutions to the equation tan x - 3tan x sin^2 x - 2 = 0 are x values such that:
1) sin x = 0 (x = 0, π, 2π, etc.)
2) tan x = 2 / (-2 + 3cos^2 x)
These are the solutions to the original equation 2tan x / (1 + tan^2 x) = sin(2x).