Question

1) sin6x+sin4x;2) cos8x-cos2x;3) tg3x+tg7x;4)ctg4x-ctgx.​

Answer 1

Ответ:

можно лучший ответ пожалуйста

Answer 2

To simplify these trigonometric expressions, you can use various trigonometric identities. Here are the simplifications for each of the given expressions:

1. sin(6x) + sin(4x):

You can use the sum-to-product trigonometric identity for sine:

sin(A) + sin(B) = 2 * sin((A + B) / 2) * cos((A - B) / 2)

In this case, A = 6x and B = 4x:

sin(6x) + sin(4x) = 2 * sin((6x + 4x) / 2) * cos((6x - 4x) / 2) sin(6x) + sin(4x) = 2 * sin(5x) * cos(x)

So, the simplified expression is:

2 * sin(5x) * cos(x)

2. cos(8x) - cos(2x):

You can use the difference-to-product trigonometric identity for cosine:

cos(A) - cos(B) = -2 * sin((A + B) / 2) * sin((A - B) / 2)

In this case, A = 8x and B = 2x:

cos(8x) - cos(2x) = -2 * sin((8x + 2x) / 2) * sin((8x - 2x) / 2) cos(8x) - cos(2x) = -2 * sin(5x) * sin(3x)

So, the simplified expression is:

-2 * sin(5x) * sin(3x)

3. tan(3x) + tan(7x):

You can use the sum-to-product trigonometric identity for tangent:

tan(A) + tan(B) = [sin(A + B)] / [cos(A) * cos(B)]

In this case, A = 3x and B = 7x:

tan(3x) + tan(7x) = [sin(3x + 7x)] / [cos(3x) * cos(7x)] tan(3x) + tan(7x) = [sin(10x)] / [cos(3x) * cos(7x)]

So, the simplified expression is:

sin(10x) / [cos(3x) * cos(7x)]

4. cot(4x) - cot(x):

You can use the difference-to-product trigonometric identity for cotangent:

cot(A) - cot(B) = [cos(A) * cos(B)] / [sin(A) * sin(B)] - [cos(B) * cos(A)] / [sin(B) * sin(A)]

Simplify by finding a common denominator:

cot(4x) - cot(x) = [cos(4x) * cos(x) - cos(x) * cos(4x)] / [sin(4x) * sin(x)]

Since the numerator cancels out, the simplified expression is:

0

So, the simplified expression for cot(4x) - cot(x) is 0.