Question

The general solution of the equation \(\tan^2 x = 1\) is

• A. \(n\pi + \frac{\pi}{4}\) only
• B. \(n\pi \pm \frac{\pi}{4}\)
• C. \(2n\pi \pm \frac{\pi}{4}\)
• D. \(n\pi - \frac{\pi}{4}\) only

Hint:

Remember the general solution formulas for squared trigonometric functions, as they are often simpler than solving the two linear cases separately:

• If \(\sin^2 x = \sin^2 \alpha\), then \(x = n\pi \pm \alpha\).

• If \(\cos^2 x = \cos^2 \alpha\), then \(x = n\pi \pm \alpha\).

• If \(\tan^2 x = \tan^2 \alpha\), then \(x = n\pi \pm \alpha\).
Notice that the format is the same for all three.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to find the general solution for the trigonometric equation \(\tan^2 x = 1\). This means finding all possible values of \(x\) that satisfy the equation.

Step 2: Key Formula or Approach:
The general solution for \(\tan^2 x = \tan^2 \alpha\) is given by the formula:
\[ x = n\pi \pm \alpha \]
where \(n\) is any integer.
First, we will find the principal value \(\alpha\) for which \(\tan^2 \alpha = 1\), and then apply the formula.

Step 3: Detailed Explanation:
The given equation is \(\tan^2 x = 1\).
We need to find an angle \(\alpha\) such that \(\tan^2 \alpha = 1\).
Taking the tangent of \(\frac{\pi}{4}\), we have \(\tan\left(\frac{\pi}{4}\right) = 1\).
Therefore, \(\tan^2\left(\frac{\pi}{4}\right) = (1)^2 = 1\).
So, we can choose our principal value \(\alpha = \frac{\pi}{4}\).
Now, we use the general solution formula for \(\tan^2 x = \tan^2 \alpha\):
\[ x = n\pi \pm \alpha \]
Substituting \(\alpha = \frac{\pi}{4}\), we get:
\[ x = n\pi \pm \frac{\pi}{4} \]
where \(n\) is any integer.

Alternative Method:
We can solve \(\tan^2 x = 1\) by taking the square root:
\[ \tan x = \pm \sqrt{1} \implies \tan x = 1 \quad \text{or} \quad \tan x = -1 \]
The general solution for \(\tan x = \tan \alpha\) is \(x = n\pi + \alpha\).
For \(\tan x = 1\), \(\alpha = \frac{\pi}{4}\). The solution is \(x = n\pi + \frac{\pi}{4}\).
For \(\tan x = -1\), \(\alpha = -\frac{\pi}{4}\). The solution is \(x = n\pi - \frac{\pi}{4}\).
Combining these two sets of solutions gives \(x = n\pi \pm \frac{\pi}{4}\).

Step 4: Final Answer:
The general solution of the equation is \(x = n\pi \pm \frac{\pi}{4}\), where \(n \in \mathbb{Z}\).