Answer:
x = {35.26°, 90°, 144.74°}
Explanation:
You want the solution to (1-cos(4x)+sin(4x))/(1+cos(4x)+sin(4x)) = 3sin(2x) making use of the given identity.
Rewrite
Using the given identity with θ = 2x, the equation becomes ...
tan(2x) = 3·sin(2x)
sin(2x)/cos(2x) = 3·sin(2x)
sin(2x)(1/cos(2x) -3) = 0
The zero product rule tells us the solutions are ...
sin(2x) = 0 ⇒ x = 90°
cos(2x) = 1/3 ⇒ x ≈ 35.26° or 144.74°
Solutions are x ∈ {35.26°, 90°, 144.74°}.
__
Additional comment
In the open interval (0, 180°), sin(2x) is 0 only for 2x = 180°, or x = 90°.
The inverse cosine of 1/3 is ±70.529°. The negative angle is an alias of 360° -70.529°. Then 2x = {70.529° or 289.471°}, giving the above result for x.
<95141404393>