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 tan²x = 1. This means finding all possible values of x that satisfy the equation.

Step 2: Key Formula or Approach (Alternate Method):
Take square root directly: tan x = ±1. Then apply general solution formula for tan x = tan α which is x = nπ + α.

Step 3: Detailed Explanation:
Given: tan²x = 1. Taking square root: tan x = 1 or tan x = -1. For tan x = 1: α = π/4 → x = nπ + π/4. For tan x = -1: α = -π/4 → x = nπ - π/4. Combine both: x = nπ ± π/4, where n is any integer.

Step 4: Final Answer:
The general solution of the equation is x = nπ ± π/4, where n ∈ Z.