Question

Sin^2(45+a)-sin^2(30-a)-sin15cos(15+2a)=sin2a 45,30,15 - градусы Cos^2(45-a) cos^2(60+a)-cos75sin(75-2a)=sin2a 75,60,45 - градусы

Answer 1

1) sin^2(45+a) - sin^2(30-a) - sin 15*cos(15+2a) =
(sin 45*cos a + cos 45*sin a)^2 - (sin 30*cos a - cos 30*sin a)^2 -
- sin 15*(cos 15*cos 2a - sin 15*sin 2a) =
= (1/√2*cos a + 1/√2*sin a)^2 - (1/2*cos a - √3/2*sin a)^2 -
- sin 15*cos 15*cos 2a + sin^2 15*sin 2a =
= 1/2*cos^2 a + 2*1/2*cos a*sin a + 1/2*sin^2 a - 1/4*cos^2 a + 
+ 2*√3/4*sin a*cos a - 3/4*sin^2 a - 1/2*sin 30*cos 2a + (1-cos 30)/2*sin 2a =
= cos^2 a*(1/2 - 1/4) + sin^2 a*(1/2 - 3/4) + sin 2a*(1/2 + √3/4 + 1/2 - √3/4) -
- 1/4*cos 2a =
= 1/4*cos^2 a - 1/4*sin^2 a + sin 2a*(1 + 0) - 1/4*cos 2a = sin 2a

2) доказывается точно также
cos^2(45-a) + cos^2(60+a) - cos 75*sin(75-2a) =
= (cos 45*cos a + sin 45*sin a)^2 + (cos 60*cos a - sin 60*sin a)^2 -
- cos(90-15)*sin(90-15-2a) =
= (1/√2*cos a + 1/√2*sin a)^2 + (1/2*cos a - √3/2*sin a)^2 -
- sin 15*cos(15+2a) =
= (1/√2*cos a + 1/√2*sin a)^2 + (1/2*cos a - √3/2*sin a)^2 -
- sin 15*(cos 15*cos 2a - sin 15*sin 2a)
Этот пример абсолютно совпадает с 1) и тоже равен sin 2a

Answer 2

To prove the given trigonometric identities, we'll start by manipulating each expression step by step. Let's tackle them one by one:

1. Proving the identity: sin^2(45 + a) - sin^2(30 - a) - sin(15)cos(15 + 2a) = sin(2a)

Step 1: Express all angles in terms of sine and cosine functions. sin(45 + a) = sin(45)cos(a) + cos(45)sin(a) = (sqrt(2)/2)cos(a) + (sqrt(2)/2)sin(a) sin(30 - a) = sin(30)cos(a) - cos(30)sin(a) = (1/2)cos(a) - (sqrt(3)/2)sin(a) sin(15) = (sqrt(2 - sqrt(3)))/2 cos(15 + 2a) = cos(15)cos(2a) - sin(15)sin(2a) = (sqrt(2 + sqrt(3)))/2 * cos(2a) - (sqrt(2 - sqrt(3)))/2 * sin(2a)

Step 2: Plug the expressions back into the original identity. sin^2(45 + a) - sin^2(30 - a) - sin(15)cos(15 + 2a) = [((sqrt(2)/2)cos(a) + (sqrt(2)/2)sin(a))^2] - [((1/2)cos(a) - (sqrt(3)/2)sin(a))^2] - [(sqrt(2 - sqrt(3)))/2 * cos(2a) - (sqrt(2 - sqrt(3)))/2 * sin(2a)] * [(sqrt(2 + sqrt(3)))/2 * cos(2a) - (sqrt(2 - sqrt(3)))/2 * sin(2a)]

Step 3: Simplify the expression using trigonometric identities. The trigonometric identities you will need are: sin^2(x) = (1 - cos(2x))/2 cos^2(x) = (1 + cos(2x))/2 sin(2x) = 2sin(x)cos(x)

After simplification, you should arrive at the identity sin(2a) on both sides of the equation.

2. Proving the identity: cos^2(45 - a) - cos^2(60 + a) - cos(75)sin(75 - 2a) = sin(2a)

Step 1: Express all angles in terms of sine and cosine functions. cos(45 - a) = cos(45)cos(a) + sin(45)sin(a) = (sqrt(2)/2)cos(a) + (sqrt(2)/2)sin(a) cos(60 + a) = cos(60)cos(a) - sin(60)sin(a) = (1/2)cos(a) - (sqrt(3)/2)sin(a) cos(75) = (sqrt(2 + sqrt(3)))/2 sin(75 - 2a) = sin(75)cos(2a) - cos(75)sin(2a) = (sqrt(2 + sqrt(3)))/2 * cos(2a) - (sqrt(2 + sqrt(3)))/2 * sin(2a)

Step 2: Plug the expressions back into the original identity. cos^2(45 - a) - cos^2(60 + a) - cos(75)sin(75 - 2a) = [((sqrt(2)/2)cos(a) + (sqrt(2)/2)sin(a))^2] - [((1/2)cos(a) - (sqrt(3)/2)sin(a))^2] - [(sqrt(2 + sqrt(3)))/2 * cos(2a) - (sqrt(2 + sqrt(3)))/2 * sin(2a)] * [(sqrt(2 + sqrt(3)))/2 * cos(2a) - (sqrt(2 + sqrt(3)))/2 * sin(2a)]

Step 3: Simplify the expression using trigonometric identities. Use the same trigonometric identities as in the previous case to simplify the expression, and you should also arrive at the identity sin(2a) on both sides of the equation.

Remember, when proving trigonometric identities, it's essential to be careful with algebraic manipulations and double-check each step for accuracy.