The number of solutions of the equation tan3x =tan x , tan x is not equal to 0 x belongs to [0,2pi] is a) 3 b) 2 c) 1 d) 0

The equation you've presented, tan(3x) = tan(x), is interesting because it involves a trigonometric function and can yield multiple solutions. To find how many solutions exist in the interval [0, 2π], we can analyze the equation step by step.

Understanding the Equation

The equation tan(3x) = tan(x) implies that the two tangent functions are equal. We can use the periodic nature of the tangent function, which has a period of π, to help us find the solutions. This equality means that the angles 3x and x differ by an integer multiple of π.

Setting Up the Equation

From tan(3x) = tan(x), we can derive the following equation:

• 3x = x + nπ, where n is any integer.

Rearranging this gives us:

• 3x - x = nπ
• 2x = nπ
• x = (nπ)/2.

Finding Values of n

Now, we need to find suitable integer values of n that keep x within the interval [0, 2π]. Let's examine the possible values of n:

• If n = 0: x = 0
• If n = 1: x = π/2
• If n = 2: x = π
• If n = 3: x = 3π/2
• If n = 4: x = 2π

These values of x are 0, π/2, π, 3π/2, and 2π, leading to five potential solutions.

Checking for Validity

However, we need to ensure that each solution does not make tan(x) equal to zero, as per the problem's constraint. The function tan(x) equals zero at the points where x = nπ, which includes 0 and π.

Thus, we must exclude the solutions:

• x = 0 (n = 0)
• x = π (n = 2)

This leaves us with the remaining valid solutions:

• x = π/2
• x = 3π/2
• x = 2π (but we can't include this either since tan(2π) = 0).

Counting the Valid Solutions

After filtering out the invalid solutions, we find that:

Thus, there are 2 valid solutions to the equation tan(3x) = tan(x) in the interval [0, 2π] where tan(x) is not equal to zero.

Final Answer

In conclusion, the number of solutions to the equation tan(3x) = tan(x) in the specified interval, ensuring tan(x) is not equal to zero, is 2. Therefore, the correct option is b) 2.