If sin⁴x + sin²x = 1, then the value of cos²x + cos⁴x is

1. 1 – 2cos⁶x
2. 1 – 2sin⁶x
3. 1
4. 0

To solve the equation sin(4x) + sin(2x) = 1 and find the value of cos(2x) + cos(4x), we can utilize some trigonometric identities and properties. Let’s break this down step by step.

Step 1: Simplifying the Given Equation

We know from trigonometric identities that the maximum value of sin functions is 1. Therefore, for sin(4x) + sin(2x) to equal 1, both terms must reach a specific combination of values. One feasible scenario is when sin(4x) = 1 and sin(2x) = 0. This occurs at certain angles.

Finding Values of x

• If sin(4x) = 1, then 4x = π/2 + 2nπ, where n is any integer.
• This leads to x = (π/8) + (nπ/2).
• For sin(2x) = 0, we have 2x = nπ, resulting in x = nπ/2.

The conditions must be satisfied simultaneously, so we look for values of x that satisfy both equations. For example, choosing n = 0 gives us x = 0, which works.

Step 2: Evaluating cos(2x) and cos(4x)

Now that we have a valid value for x, we can calculate cos(2x) and cos(4x). When x = 0:

• cos(2x) = cos(0) = 1
• cos(4x) = cos(0) = 1

Adding the Values

So, cos(2x) + cos(4x) = 1 + 1 = 2.

Step 3: Analyzing the Alternatives

If we examine the options provided in the question—1 – 2cos(6x), 1 – 2sin(6x), 1, and 0—we can check if any of them equal 2 under the same conditions. However, since we established that cos(2x) + cos(4x = 2, none of these options are equal to 2.

Final Thoughts

Ultimately, the value of cos(2x) + cos(4x) when sin(4x) + sin(2x) = 1 is clearly 2, which does not match any of the provided options. This demonstrates the importance of analyzing the trigonometric functions and their ranges. Should you have further questions about trigonometric identities or solving equations, feel free to ask!