1/2(cosa+√3sina)=cos(60˚-a)

Предыдущий
вопрос Следующий
вопрос

To prove this trigonometric identity, we can use the sum-to-product formula for cosine:

cos(A + B) = cosAcosB - sinAsinB

Let A = a and B = 60˚, and rewrite the left-hand side of the given equation using these values:

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

Now, we simplify the expression:

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

Now, we can simplify the right-hand side of the given equation by expressing it in terms of cosines:

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

Since we have shown that both sides of the equation simplify to the same expression, we have successfully proved the trigonometric identity 1/2(cos(a) + √3sin(a)) = cos(60˚ - a).