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) About Me  (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 ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-1x%2B0.125+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-1%29%5E2-4%2A1%2A0.125=0.5.

Discriminant d=0.5 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--1%2B-sqrt%28+0.5+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-1%29%2Bsqrt%28+0.5+%29%29%2F2%5C1+=+0.853553390593274
x%5B2%5D+=+%28-%28-1%29-sqrt%28+0.5+%29%29%2F2%5C1+=+0.146446609406726

Quadratic expression 1x%5E2%2B-1x%2B0.125 can be factored:
1x%5E2%2B-1x%2B0.125+=+%28x-0.853553390593274%29%2A%28x-0.146446609406726%29
Again, the answer is: 0.853553390593274, 0.146446609406726. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-1%2Ax%2B0.125+%29

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