Question

4 sin x умножить cos x sin(a+b)-cos a sin b

Answer 1

4sinxcosx = 2*2sinxcosx = 2sin2x
sin(a+b)-cosa sinb = sina cosb +sinb cosa - cosa sinb = sina cosb

Answer 2

To simplify the expression, let's break it down step by step:

1. Start with the expression: 4sin(x) * cos(x) * sin(a+b) - cos(a) * sin(b).

2. Apply the trigonometric identity: sin(a + b) = sin(a) * cos(b) + cos(a) * sin(b).

3. Substitute the identity into the expression: 4sin(x) * cos(x) * (sin(a) * cos(b) + cos(a) * sin(b)) - cos(a) * sin(b).

4. Distribute the terms: (4sin(x) * cos(x) * sin(a) * cos(b)) + (4sin(x) * cos(x) * cos(a) * sin(b)) - cos(a) * sin(b).

5. Apply the trigonometric identity: sin(2θ) = 2sin(θ) * cos(θ).

6. Simplify the expression using the identity: 2sin(x) * cos(x) = sin(2x).

7. Apply the identity to the first two terms: (sin(2x) * sin(a) * cos(b)) + (sin(2x) * cos(a) * sin(b)) - cos(a) * sin(b).

8. Rearrange the terms: sin(2x) * sin(a) * cos(b) + sin(2x) * cos(a) * sin(b) - cos(a) * sin(b).

9. Factor out sin(b): sin(2x) * sin(a) * cos(b) + sin(b) * (sin(2x) * cos(a) - cos(a)).

That is the simplified form of the expression: sin(2x) * sin(a) * cos(b) + sin(b) * (sin(2x) * cos(a) - cos(a)).