Question

cos(a) - cos(60-a)=cos(60+a) Помогите доказать, что равенство верно! Заранее спасибо)

Answer 1

cos(a) - cos(60-a)=cosa-cosacos60-sin60sina=cosa(1-cos60)-sin60sina=cosacos60-sinasin60=cos(60+a)

Answer 2

Proof of the Equation cos(a) - cos(60-a) = cos(60+a)

To prove the equation cos(a) - cos(60-a) = cos(60+a), we can use the trigonometric identity:

cos(a) - cos(b) = -2 * sin((a+b)/2) * sin((a-b)/2)

Applying this identity to the equation, we have:

cos(a) - cos(60-a) = -2 * sin((a+(60-a))/2) * sin((a-(60-a))/2)

Simplifying further:

cos(a) - cos(60-a) = -2 * sin((a+60-a)/2) * sin((a-60+a)/2)

cos(a) - cos(60-a) = -2 * sin(60/2) * sin(2a-60/2)

cos(a) - cos(60-a) = -2 * sin(30) * sin(a-30)

Since sin(30) = 1/2, we can substitute it into the equation:

cos(a) - cos(60-a) = -2 * (1/2) * sin(a-30)

Simplifying further:

cos(a) - cos(60-a) = -sin(a-30)

Now, let's simplify the right side of the equation:

cos(60+a) = cos(60) * cos(a) - sin(60) * sin(a)

Since cos(60) = 1/2 and sin(60) = √3/2, we can substitute these values into the equation:

cos(60+a) = (1/2) * cos(a) - (√3/2) * sin(a)

Now, let's simplify the left side of the equation:

cos(a) - cos(60-a) = -sin(a-30)

cos(a) - cos(60-a) = -sin(a) * cos(30) + cos(a) * sin(30)

Since cos(30) = √3/2 and sin(30) = 1/2, we can substitute these values into the equation:

cos(a) - cos(60-a) = -sin(a) * (√3/2) + cos(a) * (1/2)

Simplifying further:

cos(a) - cos(60-a) = (-√3/2) * sin(a) + (1/2) * cos(a)

Comparing the right side of the equation with the expression for cos(60+a), we can see that they are equal:

-sin(a) * (√3/2) + cos(a) * (1/2) = (1/2) * cos(a) - (√3/2) * sin(a)

Therefore, we have proved that the equation cos(a) - cos(60-a) = cos(60+a) is true.

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