Question

A) sin3x+sinx-sin2x = 2cosx(cosx-1)

Answer 1

решение на фото внизу

Answer 2

Let's prove the given trigonometric identity step by step. The identity to be proven is:

A) sin(3x) + sin(x) - sin(2x) = 2cos(x)(cos(x) - 1)

We'll use various trigonometric identities to simplify the left-hand side (LHS) and the right-hand side (RHS) and then show that they are equal. Here's the proof:

Step 1: Express sin(3x) using the angle addition formula for sine: sin(3x) = sin(2x + x) = sin(2x)cos(x) + cos(2x)sin(x)

Step 2: Express sin(2x) using the angle double formula for sine: sin(2x) = 2sin(x)cos(x)

Step 3: Substitute the value of sin(2x) into the expression for sin(3x): sin(3x) = 2sin(x)cos(x)cos(x) + cos(2x)sin(x)

Step 4: Express cos(2x) using the double angle formula for cosine: cos(2x) = cos^2(x) - sin^2(x)

Step 5: Substitute the value of cos(2x) into the expression for sin(3x): sin(3x) = 2sin(x)cos(x)cos(x) + (cos^2(x) - sin^2(x))sin(x)

Step 6: Simplify the expression for sin(3x): sin(3x) = 2sin(x)cos^2(x) + cos^2(x)sin(x) - sin^2(x)sin(x)

Step 7: Combine like terms: sin(3x) = 3sin(x)cos^2(x) - sin^2(x)sin(x)

Step 8: Factor out sin(x) from the right-hand side (RHS): sin(3x) = sin(x)(3cos^2(x) - sin^2(x))

Step 9: Apply the double angle formula for cosine: cos(2x) = 2cos^2(x) - 1

Step 10: Substitute the value of cos(2x) into the expression for sin(3x): sin(3x) = sin(x)(3cos^2(x) - sin^2(x)) = sin(x)(2cos^2(x) - 1)

Step 11: Combine the terms on the left-hand side (LHS): LHS = sin(3x) + sin(x) - sin(2x) = sin(x)(2cos^2(x) - 1) + sin(x) - 2sin(x)cos(x)

Step 12: Factor out sin(x) from the LHS: LHS = sin(x)(2cos^2(x) - 1 + 1 - 2cos(x))

Step 13: Simplify the expression in parentheses on the LHS: LHS = sin(x)(2cos^2(x) - 2cos(x))

Step 14: Factor out 2cos(x) from the expression on the right-hand side (RHS): RHS = 2cos(x)(cos(x) - 1)

Step 15: The left-hand side (LHS) and the right-hand side (RHS) are equal: LHS = 2cos(x)(cos(x) - 1) = RHS

Thus, we have proved the given trigonometric identity A) sin(3x) + sin(x) - sin(2x) = 2cos(x)(cos(x) - 1).