tan^2x=tanx+2

Note that the equation can be rearranged and factored as follows:

tan^2x -tanx -2 = 0
(tanx -2)(tanx +1) = 0
tan x = 2 or -1
There are four angles between 0 and 360 degrees that are solutions. One of them is 135 degrees. Another is 63.435 degrees. See if you can find the others.

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To solve the equation tan^2x = tanx + 2, we can rewrite it as a quadratic equation by moving all the terms to one side:

tan^2x - tanx - 2 = 0

Now, let's substitute tanx with y:
y^2 - y - 2 = 0

This is now a quadratic equation in the variable y. To solve it, we can factor it or use the quadratic formula.

If we attempt to factor the equation, we need to find two numbers that multiply to -2 and add up to -1. The numbers that satisfy these conditions are -2 and +1:
(y - 2)(y + 1) = 0

Setting each factor equal to zero, we get two separate equations:
y - 2 = 0
y + 1 = 0

Solving these equations will give us the possible values of y:
y = 2
y = -1

Now, remember that we substituted tanx for y. So, we need to substitute back tanx for y in order to find the values for x:
tanx = 2
tanx = -1

To solve for x, we need to take the inverse tangent (arctan) of both sides of the equation:
x = arctan(2)
x = arctan(-1)

Now, we need to consider the range of values for x. The tangent function has a period of pi, which means we can add or subtract multiples of pi and still get the same value of tangent. Therefore, we should consider the values of x in the given range.

Using a calculator, we can approximate the values of x:
x ≈ 1.107 + n * pi
x ≈ -0.785 + n * pi

Here, n is an integer that can take any positive, negative, or zero value.

So, the solutions to the equation tan^2x = tanx + 2, within the given range, are:
x ≈ 1.107 + n * pi
x ≈ -0.785 + n * pi

Note that n * pi accounts for all possible values of tangent within the given range.