SOLUTION: Solve the equation. tan(x)+cot(x)=4sin(2x)

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Question 1032256: Solve the equation.

tan(x)+cot(x)=4sin(2x)
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
tan(x)+cot(x)=4sin(2x)
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tan(x)+cot(x) = 8sin(x)*cos(x)
Multiply thru by sin*cos
sin^2 + cos^2 = 8sin^2*cos^2
8sin^2cos^2 = 1
sin^2*(1 - sin^2) = 1/8
sin^4 - sin^2 + 1/8 = 0
Sub x for sin^2
x^2 - x + 1/8 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation (in our case ) has the following solutons:



For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=0.5 is greater than zero. That means that there are two solutions: .




Quadratic expression can be factored:

Again, the answer is: 0.853553390593274, 0.146446609406726. Here's your graph:

=============
sin^2 =~ 0.853553390593274
x = 67.5, 112.5 + k*360 degs
x = 3pi/8, 5pi/8 + k*2pi radians
k = any integer
===================
sin^2 = 0.146446609406726 --> complements of angles above.
===============


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