SOLUTION: cos(2\alpha +(\pi )/(4))=3tan(2\alpha +(\pi )/(4))

I read this equation in this way

     = .


Notice that the argument under cosine function is the same as under the tangent function.

So, I introduce new variable

    x = .


Then the given equation takes the form

    cos(x) = 3*tan(x).


It implies

    cos^2(x) = 3*sin(x),

    1-sin^2(x) = 3*sin(x)

    sin^2(x) + 3*sin(x) - 1 = 0

It is a quadratic equation relative sin(x).  Use the quadratic formula

    sin(x) =  = .


Since sin(x) must be between -1 and 1, only the root sin(x) =  = 0.30278  really fits.


Hence,  there are two solutions for x:

    x = arcsin(0.30278) = 0.3076  OR  x =  = 3.14159 - 0.3076 = 2.834 (rounded).


It gives two solutions for :

    (a)   =  =  = -0.2389  (rounded);

    (b)   =   =   =  1.0243  (rounded).