Sin6xcos5x=sin14xcos13x

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To verify this trigonometric identity, we can use the double-angle and half-angle formulas for sine and cosine.

Starting with the left side of the equation:
sin(6x)cos(5x)

Using the double-angle formula for sine, we have:
sin(6x)cos(5x) = 2sin(3x)cos(3x)cos(5x)

Applying the double-angle formula for cosine, we get:
2sin(3x)cos(3x)cos(5x) = sin(6x)sin(8x)

Expanding the right side of the equation:
sin(14x)cos(13x)

Using the double-angle formula for sine:
sin(14x)cos(13x) = 2sin(7x)cos(7x)cos(13x)

Again, applying the double-angle formula for cosine:
2sin(7x)cos(7x)cos(13x) = sin(14x)sin(26x)

Now we compare the left and right sides of the equations:
sin(6x)sin(8x) = sin(14x)sin(26x)

Since the angles on both sides of the equation are the same, we can simplify further using the angle addition formula for sine:
sin(a)sin(b) = 1/2[cos(a - b) - cos(a + b)]

Thus:
sin(6x)sin(8x) = 1/2[cos(-2x) - cos(20x)]

Since the cosine function is even:
1/2[cos(-2x) - cos(20x)] = 1/2[cos(2x) - cos(20x)]

Which is equal to:
sin(14x)sin(26x)

Therefore, the trigonometric identity sin(6x)cos(5x) = sin(14x)cos(13x) is verified.