Question

4√(3)cosx-4sinx=2√(3)cos^2x-sin2x

Answer 1

Ответ в прикрепленном файле

Answer 2

To solve the equation 4√(3)cosx - 4sinx = 2√(3)cos^2x - sin2x, we'll start by simplifying and rearranging the terms.

Let's focus on the right side of the equation first:

2√(3)cos^2x - sin2x

Using the double-angle formula for sine, sin2x = 2sinxcosx, we can rewrite the equation as:

2√(3)cos^2x - 2sinxcosx

Factoring out a common term of 2cosx gives:

2cosx(√(3)cosx - sinx)

Now, let's go back to the original equation:

4√(3)cosx - 4sinx = 2cosx(√(3)cosx - sinx)

Dividing both sides of the equation by 2cosx, we get:

2√(3) - 2tanx = √(3)cosx - sinx

Now, let's rearrange the terms:

√(3)cosx - sinx - 2√(3) + 2tanx = 0

Next, we can combine like terms:

(√(3)cosx - sinx) + 2(tanx - √(3)) = 0

Now, let's use the identity tanx = sinx/cosx:

(√(3)cosx - sinx) + 2(sinxcosx - √(3)cosx) = 0

Expanding the terms gives:

√(3)cosx - sinx + 2sinxcosx - 2√(3)cosx = 0

Combining like terms:

(√(3) - 2√(3))cosx + (2sinxcosx - sinx) = 0

Simplifying:

-√(3)cosx + (2sinxcosx - sinx) = 0

Now, let's factor out common terms:

cosx(-√(3) + 2sinx) - sinx = 0

Finally, we have the simplified form:

cosx(-√(3) + 2sinx) - sinx = 0

This equation represents the relationship between x, and further simplification is not possible without additional context or constraints.