Solve from [0,2pi).
cot^2(4x)tan(2x)-tan(2x)=0
How do you solve this (step-by-step)?
first take out a common factor
tan2x(cot^2(4x) - 1) = 0
tan2x(cot4x + 1)(cot4x - 1) = 0
tan2x = 0 OR cot4x = -1 OR cot4x = 1
so if
tan2x = 0
2x = 0 or pi
x = 0 or pi/2
if cot4x = 1
tan 4x = 1
4x = pi/4 or 4x = 5pi/4
x = pi/16 or x = 5pi/16
if cot4x = -1
4x = pi-pi/4 or 4x = 2pi-pi/4
x = 3pi/16 or x = 7pi/16
so x = 0, pi/16, 3pi/16, 5pi/16, 7pi/16
Thank you so much!
To solve the equation cot^2(4x)tan(2x) - tan(2x) = 0 on the interval [0, 2π), you can follow these step-by-step instructions:
Step 1: Factor out the common term. In this case, tan(2x) is a common term, so we can factor it out:
tan(2x)(cot^2(4x) - 1) = 0
Step 2: Apply the zero-product property. The zero-product property states that if the product of two factors is equal to zero, then at least one of the factors must be equal to zero. So, each factor must be set equal to zero and solved individually.
Consider the first factor: tan(2x) = 0
To find the solutions for this equation, we need to find the values of 2x that make the tangent equal to zero. The tangent function is zero at x = 0, π (180 degrees), 2π (360 degrees), and so on. However, we are working on the interval [0, 2π), so the solutions within this interval are x = 0 and x = π.
Now consider the second factor: cot^2(4x) - 1 = 0
First, simplify the equation by adding 1 to both sides:
cot^2(4x) = 1
Step 3: Solve for cot(4x). To find the values of 4x that make the cotangent equal to ±1, we can use the properties of the cotangent function. The cotangent is equal to 1 when the angle is π/4 (45 degrees), and it is equal to -1 when the angle is 5π/4 (225 degrees). Thus, we have two cases to consider.
Case 1: cot(4x) = 1
To find the solutions within the given interval, we can solve 4x = π/4, which gives x = π/16.
Case 2: cot(4x) = -1
To find the solutions within the given interval, we can solve 4x = 5π/4, which gives x = 5π/16.
So, the solutions to the equation cot^2(4x)tan(2x) - tan(2x) = 0 on the interval [0, 2π) are x = 0, x = π, x = π/16, and x = 5π/16.