To prove this equation, we need to use the sum-to-product trigonometric identity, which states that:
cos(a)cos(b) = 0.5[cos(a+b) + cos(a-b)]
Therefore, we can rewrite the left side of the equation as:
cos(x)cos(3x) = 0.5[cos(4x) + cos(2x)]
Next, we can rewrite the right side of the equation as:
cos(5x)cos(7x) = 0.5[cos(12x) + cos(-2x)]
Since cos(-θ) = cos(θ), we have:
cos(-2x) = cos(2x)
Therefore, the right side can be simplified as:
cos(5x)cos(7x) = 0.5[cos(12x) + cos(2x)]
Now, we can see that both sides of the equation are equal to 0.5[cos(12x) + cos(2x)], which means:
cos(x)cos(3x) = cos(5x)cos(7x)
Therefore, the equation is proven to be true.
To prove this equation, we need to use the sum-to-product trigonometric identity, which states that:
cos(a)cos(b) = 0.5[cos(a+b) + cos(a-b)]
Therefore, we can rewrite the left side of the equation as:
cos(x)cos(3x) = 0.5[cos(4x) + cos(2x)]
Next, we can rewrite the right side of the equation as:
cos(5x)cos(7x) = 0.5[cos(12x) + cos(-2x)]
Since cos(-θ) = cos(θ), we have:
cos(-2x) = cos(2x)
Therefore, the right side can be simplified as:
cos(5x)cos(7x) = 0.5[cos(12x) + cos(2x)]
Now, we can see that both sides of the equation are equal to 0.5[cos(12x) + cos(2x)], which means:
cos(x)cos(3x) = cos(5x)cos(7x)
Therefore, the equation is proven to be true.