Question

Сократить дробь: (sin5x + sin3x)/(cos6x*sin10x-cos10x*sin6x)

Answer 1

$\frac{\sin5x+\sin3x}{\sin10x\cos6x-\cos10x\sin6x}=\frac{2\sin\left(\frac{5x+3x}2\right)\sin\left(\frac{5x-3x}2\right)}{\sin(10x-6x)}=\frac{2\sin4x\sin x}{\sin4x}=2\sin x$

Answer 2

To simplify the given expression, we can start by using trigonometric identities to simplify the numerator and denominator separately. Let's begin with the numerator:

Numerator: sin(5x) + sin(3x)

We can use the trigonometric identity: sin(A) + sin(B) = 2*sin((A + B)/2) * cos((A - B)/2)

Applying this identity to the numerator:

sin(5x) + sin(3x) = 2 * sin((5x + 3x)/2) * cos((5x - 3x)/2) = 2 * sin(4x) * cos(x)

Next, let's simplify the denominator:

Denominator: cos(6x) * sin(10x) - cos(10x) * sin(6x)

We can use the trigonometric identity: cos(A) * sin(B) - cos(B) * sin(A) = sin(B - A)

Applying this identity to the denominator:

cos(6x) * sin(10x) - cos(10x) * sin(6x) = sin(10x - 6x) = sin(4x)

Now, our simplified expression becomes:

(2 * sin(4x) * cos(x)) / sin(4x)

Now, we can further simplify the expression by canceling out the common factor of sin(4x) from the numerator and denominator:

= 2 * cos(x)

So, the simplified expression is 2 * cos(x).