## 1 + tan^2x =

### To solve the equation 1 + tan^2x = 0, we can start by manipulating the equation using trigonometric identities

To solve the equation 1 + tan^2x = 0, we can start by manipulating the equation using trigonometric identities.

The trigonometric identity that relates tan^2x to other trigonometric functions is:

tan^2x + 1 = sec^2x

Using this identity, we can rewrite the equation as:

tan^2x + 1 = 0

Now, let’s isolate the term tan^2x by subtracting 1 from both sides:

tan^2x = -1

Next, we can take the square root of both sides:

√(tan^2x) = ±√(-1)

Since the square root of a negative number is not a real number, there are no solutions to this equation in the real number system.

However, in the complex number system, the square root of -1 can be denoted by the imaginary unit i. In this case, we can introduce the concept of imaginary numbers and state that tan^2x = -1 has imaginary solutions.

Therefore, the equation 1 + tan^2x = 0 has no real number solutions, but it has imaginary solutions.

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